Magnetic-field evolution with large-scale velocity circulation in a neutron-star crust

The equation of bulk flow motion is given by

where $\rho$ represents the mass density, and ${\vec{f}}$ is the force per unit volume. The force density ${\vec{f}}$ in the crust comprises pressure, gravity, the Lorentz force, and the plastic-flow term. The deformation is elastic below a threshold, but we ignore the regime in this paper. We assume an incompressible fluid for the sake of simplicity and explicitly consider

where $\nu$ is the viscosity coefficient. The last term in eq.(6) is physically the same as a standard viscous-flow term (Lander, 2016). Based on the condition ${\vec{\nabla}}\cdot{\vec{v}}_{\rm b}=0$, the last term may be written as

It is impossible to simultaneously follow the bulk flow motion given by eq.(5), when the long-term evolution of the magnetic fields is calculated. The order of their timescales is extremely different. As our relevant timescale is much longer than the dynamical timescale, we approximate ${\vec{f}}=0$. That is, the forces are always balanced for a short timescale, such that the velocity-field structure is subjected to an equilibrium state in a moment.

Another difficult problem arises in calculating the velocity field based on the force balance ${\vec{f}}=0$. There are four types of forces involved in eq.(6), but their magnitudes are quite different. The degenerate pressure and gravity represented by eq.(6) are dominant terms, and they actually determine the stellar structure. The barotropic structure is generally a good approximation. Thus, the first and second terms are regarded as irrotational vectors. It should be noted that any vector can be decomposed by using an irrotational vector and a solenoidal vector (Helmholtz decomposition, see Arfken & Weber, 2005). The Lorentz force comprises a mixture of the irrotational and solenoidal vectors. The solenoidal part produced during the magnetic-field evolution has no relation to the dominant forces, i.e., pressure and gravity. We herein assume that the bulk motion arises from the solenoidal part of the Lorentz force. Thus, a small component of the magnitude is extracted by the vectorial property. The velocity ${\vec{v}}_{\rm b}$ is assumed to be determined by

Our concern in terms of this paper is how a new component ${\vec{v}}_{\rm b}$ in eq.(2) will change the magnetic-field evolution given by eq.(1). We explicitly describe how the bulk velocity ${\vec{v}}_{\rm b}$ is determined in an axially symmetric case. A general form of the incompressible flow is given by two functions $F$ and $H$

where ${\vec{e}}_{\phi}$ is a unit vector in the $\phi$ direction, and $\varpi=r\sin\theta$ in spherical coordinates $(r,\theta,\phi)$. The vorticity is calculated as

Here, a new function $W$ is introduced by $F$,

This equation is explicitly written as

Based on the assumption of the axially symmetric configuration, $f_{\phi}=0$ is reduced to

Equation (8) for $W$ is reduced to

where the conditions ${\vec{\nabla}}\cdot{\vec{j}}={\vec{\nabla}}\cdot{\vec{B}}=0$ are used. Thus, the bulk velocity field is determined by solving two second-order partial differential equations (13) and (14) of an elliptic type, and the stream function $F$ of the velocity in eq.(9) is determined by solving eq.(11).

Numerical methods for describing the magnetic-field evolution are discussed(Pons & Viganò, 2019, for a review and references). There are several choices of variables, but we calculate the time-integration of $B_{\phi}$ and $j_{\phi}$ in this study. From eq.(1) and on applying a rotation to it, we obtain a set of evolution equations for $B_{\phi}$ and $j_{\phi}$

and

Once $B_{\phi}$ and $j_{\phi}$ are integrated with respect to time, the poloidal current ${\vec{j}}_{\rm p}$ and poloidal magnetic field ${\vec{B}}_{\rm p}$ at each time are determined using

It is convenient to solve the last equation by introducing a magnetic function $\Psi$ that satisfies

Equation (18) is an elliptical equation of $\Psi$,

Instead of solving eq.(20) at each time step, another method comprises the time-integration of $\Psi$. From eqs.(1) and (16), the evolution of $\Psi$ is given by the azimuthal component of the electric field,

We adopt eq.(21) as a numerical method, as it is found to be accurate and fast in our numerical calculations.

It is instructive to write down energy-conservation equation¹¹1The meridional flow will have associated changes in chemical composition and may thus trigger e.g. electron capture reactions that might generate entropy. However, these are likely points for future research. We here consider the magnetic-field evolution in a fixed structure.. The magnetic energy inside a crust of volume $V$ is given by

and its time derivative is given by

The first term represents the Joule loss, while the second represents the work done by the Lorentz force inside the crust. For the Hall evolution, i.e., the velocity in eq.(3) is given by ${\vec{v}}=-{\vec{j}}/({\rm e}n_{\rm e})\equiv{\vec{v}}_{\rm H}$ such that the work vanishes. The third term in eq.(23) represents the Poynting energy lost through a surface. We found that this term is very small in our secular simulation of magnetic fields.

We solve a set of time-dependent equations for $B_{\phi}$, $j_{\phi}$, and $\Psi$, and a set of elliptic-type equations for the bulk velocity fields $W$, $F$, and $H$ inside a crust. The spherical shell region of the crust is represented by $r_{c}\leq r\leq R$ and $0\leq\theta\leq\pi$. We discuss the boundary conditions for these functions.

The boundary condition at the symmetric axis ($\theta=0,\pi$) is a regularity condition. That is, all the functions $B_{\phi}$, $j_{\phi}$, $\Psi$, $W$, $F$, and $H$ should vanish at the axis.

We assume that the magnetic fields are expelled from the inner core and that they are maintained by electric currents flowing in the crust. This assumption simplifies the problem, as core magnetic fields do not affect crustal fields. We impose the conditions $B_{\phi}=j_{\phi}=\Psi=0$ for the magnetic fields. Based on eqs.(17) and (19), these conditions indicate that no radial components of the current and magnetic field cross the boundary at $r_{c}$. In the case of the velocities, we impose the conditions $W=F=H=0$, that is, the bulk flow is suppressed at the interface of the core.

Finally, we discuss the conditions at the surface $r=R$. The toroidal magnetic field and electric current are confined inside a star, such that we impose $B_{\phi}=j_{\phi}=0$. The poloidal magnetic field is not confined inside the star but is continuously connected to the exterior vacuum solution. We impose the continuity of the magnetic function $\Psi$ as the boundary condition. This indicates that the continuity of $B_{r}$ and discontinuity of a tangential component are generally allowed by the surface current. In the case of the bulk motions, the fixed boundary conditions $W=F=H=0$ are imposed. These are the same as those at $r_{c}$.

Our consideration is limited to the inner crust of a neutron star, where the mass density ranges from $\rho_{c}=1.4\times 10^{14}$ g cm^-3 at the core-crust boundary $r_{c}$ to $\rho_{1}=4\times 10^{11}$ g cm^-3 at the neutron-drip point $R$. We ignore the outer crust, the structure of which may be significantly affected in the presence of a strong magnetic field, and we treat the exterior region of $r>R$ as the vacuum. The crust thickness is assumed to be $\Delta r/R=(R-r_{c})/R=0.1$.

The density profile of $r_{c}\leq r\leq R$ is approximated as follows (Lander & Gourgouliatos, 2019).

Using the normalized density ${\hat{\rho}}$, the distribution of the electron number density $n_{\rm e}$, electric conductivity $\sigma$, and viscous coefficient $\nu$ are approximated using analytical expressions(Lander & Gourgouliatos, 2019).

The numerical coefficients in front of these equations are

We consider a simple form of electric conductivity $\sigma\propto n_{\rm e}^{3/2}$, but the coefficient $\sigma_{c}$ may change by an order of magnitude in the actual situation, as $\sigma_{c}$ depends on the thermal evolution of the neutron star. Using these numerical values, we estimate the typical timescales, the Ohmic timescale $\tau_{\rm Ohm}$, Hall timescale $\tau_{\rm H}$, and bulk motion timescale $\tau_{\rm v}$ as follows.

Herein, we used the maximum values for $n_{\rm e}$, $\sigma$, and $\nu$, i.e., the values at the core–crust boundary. The magnetic field in our model is zero at the boundary, such that the volume averaged value $B_{\rm av}$ is used in these estimates. It should be noted that the actual timescales vary substantially. In particular, $n_{\rm e}$ decreases by a factor of $10^{-2}$ and $\nu$ by a factor of $10^{-3}$ at the surface $r=R$. The configuration near the surface changes in much smaller timescales than the above ones. The timescale $\tau_{\rm v}$ is relevant to our quasi-stationary approximation. In a certain timescale $\ll\tau_{\rm v}$, all the forces are assumed to be balanced, and the bulk velocity is determined. When $\tau_{\rm v}$ becomes too small for some parameters, the quasi-stationary approximation may not be justified, and we have to take into consideration microscopic dynamics.

The following relation by order-of-magnitude may be useful for understanding the effect of the bulk motion in the numerical simulation.

where $v_{\rm H}$ is the Hall velocity that is given by the second term in eq.(3). Equation (34) indicates that the bulk velocity $v_{\rm b}$ increases as the viscosity decreases. It should also be noted that both $v_{\rm b}$ and $v_{\rm H}$ increase as the magnetic field strength increases, but the ratio $v_{\rm b}/v_{\rm H}$ increases. Thus, the bulk velocity induced by the Lorentz force is important in low-viscosity and strong-magnetic-field regimes.

We discuss the initial model of the magnetic fields. A reasonable state comprises a barotropic MHD equilibrium (Gourgouliatos et al., 2013). We assume that the toroidal field vanishes, i.e., $B_{\phi}=0$. The azimuthal component of the electric current $j_{\phi}$ for the barotropic MHD equilibrium is given as follows(Gourgouliatos et al., 2013).

where $f_{\rm B}$ is generally a function of $\Psi$ only, and $f_{\rm B}$ is assumed to be a constant for the sake of simplicity. Based on eqs.(19) and (20), the magnetic function $\Psi$ and poloidal magnetic field ${\vec{B}}_{\rm p}$ are calculated. The functional form (35) with a constant $f_{\rm B}$ indicates that the poloidal field is given by the dipole component $l=1$ only when we expand $\Psi$ with Legendre polynomials as $\Psi=-\sum g_{l}\sin\theta\partial P_{l}/\partial\theta$.

The magnetic fields in the MHD equilibrium are not fixed in a longer timescale. Their state is not in equilibrium for the electrons and is therefore evolved. In order to explain this, we neglect the Joule term and bulk motion in eqs.(1) and (2). The initial Lorentz force for eq.(35) is given by ${\vec{j}}\times{\vec{B}}=j_{\phi}{\vec{\nabla}}\Psi/\varpi$ $=\rho f_{\rm B}{\vec{\nabla}}\Psi$ and the evolution of the magnetic field at the beginning is given by

where $\chi\equiv\rho f_{\rm B}/({\rm e}n_{\rm e})$ is proportional to the inverse of the electron number fraction. In our model given by eq.(25), $\chi$ depends on the radial position, and hence, the magnetic field evolution is driven by the Hall term.

In Fig. 1, we present two models in MHD equilibrium. In order to display the detailed structure in the crust, the region $(r\sin\theta,r\cos\theta)$, ($0.9\leq r/R\leq 1$) is enlarged by five times in the figure as $(\xi(r)\sin\theta,\xi(r)\cos\theta)$, where $\xi=1+(R-r)/(2\Delta r)$. The magnetic field in both the models is matched at the surface to the external dipole field of the same strength, but there is a difference in the surface current distribution at $r=R$. In model A shown in the left panel of Fig. 1, the surface current is allowed, such that $B_{\theta}$ is discontinuous. In model B shown in the middle panel of Fig. 1, there is no surface current, such that $B_{\theta}$ is continuously connected to the external field.

As shown in Fig. 1, there is a stronger current flowing in model B than that in model A. As a result, the internal field of model B is stronger than that of model A. In order to compare their field strength quantitatively, we define a volume-average of the magnetic field using the root-mean-square value.

The numerical calculation shows that the average strength is $B_{\rm av}=4.3B_{0}$ for model A and $B_{\rm av}=6.7B_{0}$ for model B, where $B_{0}$ is the magnetic-field strength at the pole $(r=R,\theta=0)$. The magnetic energy $E_{\rm B}$ stored in the crust part is $E_{\rm B}=0.85B_{0}^{2}R^{3}$ for model A and $E_{\rm B}=2.1B_{0}^{2}R^{3}$ for model B. Almost twice the energy of model A is deposited in model B by the strong currents.

As discussed in subsection 3.1, the magnetic fields in the MHD equilibrium are not fixed, but change according to the Hall drift. The magnetic energy decays by the Joule loss on a secular timescale. We calculated the time evolution for two initial models described in the previous subsection and compared the results as shown in Fig. 2. The numerical energy conservation for the initial model B is also indicated by a dotted line. The sum of the magnetic energy and time-integrated Joule-loss is almost constant. The results for model B are indicated by thick curves. The magnetic energy $E_{\rm B}$ decreases by half at the time $\sim 0.3\tau_{\rm H}$. The results for model A are also indicated by thin lines, but their variations are quite slow. The decrease is given by $\Delta E_{\rm B}=0.01B_{0}^{2}R^{3}$ at $\tau_{\rm H}\sim 0.12\tau_{\rm Ohm}$. It takes a longer time $\sim\tau_{\rm Ohm}$ to observe an evident magnetic decay in model A. In model B, a larger amount of current is initially confined to the inner region and then drifts to the outer high-resistivity region and dissipates in a shorter timescale $\sim 0.3\tau_{\rm H}\sim 0.04\tau_{\rm Ohm}$.

The numerical simulation with the Joule and Hall terms is characterized by the so-called magnetic Reynolds number or a magnetization parameter, $B\sigma/({\rm e}cn_{\rm e})$, which is a ratio of the Joule to Hall timescales (see eqs. (31)–(32).) As an overall normalization of Fig. 2, we take $B_{0}=5\times 10^{13}$G ($\sim B_{\rm Q}\equiv 4.4\times 10^{13}$G), which is a marginal value between that of magnetars and pulsars. The average strength in the interior for model B is $B_{\rm av}\approx 3\times 10^{14}$G, the initial magnetic energy is $E_{\rm B}\approx 6\times 10^{45}$erg, and its half-life corresponds to $0.3\tau_{\rm H}\sim 0.04\tau_{\rm Ohm}\sim 2\times 10^{5}$yr. It is evident that the magnetic energy increases $E_{\rm B}\propto B_{0}^{2}$ and the evolution timescale $\tau_{\rm H}\propto B_{0}^{-1}$ becomes short for magnetars with $B_{0}>5\times 10^{13}$G. Furthermore, the dissipation rate ($\propto E_{\rm B}\tau_{\rm H}^{-1}\propto B_{0}^{3}$) significantly depends on the magnetic field strength $B_{0}$.

The magnetic energy decreases in a timescale $\sim\tau_{\rm H}$, but the external field is retained as a dipole with the initial field strength. This is related to the choice of the initial model. We adopt a smooth configuration, in which the poloidal field is dipolar and toroidal field is absent. The coupling between the poloidal and toroidal fields is suppressed, although the toroidal field is generally produced. In the next subsection, we incorporate a viscous bulk flow in this simple system, and the resulting effects are easily understood.

Figure 3 shows the magnetic energy evolution with bulk motion for the initial magnetic field given by model B. On taking into consideration the uncertainty of the viscosity coefficient, we calculated three models by changing $\nu$ while using the same spatial profile. The fiducial value $\nu_{c}$ is given by eq.(30), and the magnitude is changed by a factor of 5, i.e., models with $0.2\nu_{c}$, $\nu_{c}$, and $5\nu_{c}$ are calculated.

On including the bulk motions, the magnetic energy is transferred to two channels: heat via Joule loss and bulk kinetic energy via work (see eq.(23)). The latter initially acts as the magnetic energy loss, as the bulk velocity is set as zero. Thus, the initial decrease in the magnetic energy increases in speed owing to the bulk motion. Half the magnetic energy is dissipated at the time $0.15\tau_{\rm H}$ in the fiducial model. The Joule loss and work rate are comparable in magnitude at the early stage $\sim 0.05\tau_{\rm H}$. Both of these tend to be ineffective at approximately $0.3\tau_{\rm H}$. Till this time, the magnetic energy decrease is $\Delta E_{\rm B}=1.22B_{0}^{2}R^{3}$, the integrated Joule loss energy is $0.88B_{0}^{2}R^{3}$, and the net work is $0.34B_{0}^{2}R^{3}$. On comparing these with results obtained without considering bulk motion, it is determined that a larger amount of magnetic energy is lost by $\sim$4% in the fiducial model. On reducing the viscosity, this effect is enhanced, and the value becomes 16% in the low-viscosity model with $0.2\nu_{c}$. The energy loss is enhanced not by the Joule heating but by the work performed. That is, the magnetic energy is used to drive the bulk motions.

Figure 4 shows the fluid motion for the fiducial viscosity at the time $0.02\tau_{\rm H}$. In the left panel, the contours of a stream function $F$ are presented, while those of the azimuthal velocity $v_{\phi}$ are shown in terms of an “enlarged" coordinate. The stream function $F$ is anti-symmetric with respect to $\theta=\pi/2$, that is, $F<0$ in the upper region ($0<\theta<\pi/2$), but $F>0$ in the lower region ($\pi/2<\theta<\pi$) in Fig. 4. Thus, the meridional circulation flow is counter-clockwise in the upper region, while the flow becomes clockwise. The maximum of $|v_{\theta}|$ occurs at $\theta\approx 0.2\pi$ and $\theta\approx 0.8\pi$ in the vicinity of the surface: $v_{\theta}\approx-7$cm yr^-1 at $\theta\approx 0.2\pi$ and $v_{\theta}\approx+7$cm yr^-1 at $\theta\approx 0.8\pi$. The radial component $|v_{r}|$ is one order of magnitude smaller than $|v_{\theta}|$. Using eq.(9), we have a relation between them, $|v_{r}|=|\varpi^{-1}\nabla_{\theta}F|$ $\ll|v_{\theta}|=|\varpi^{-1}\nabla_{r}F|$, wherein the inequality can be attributed to a steep slope of $F$ in the radial direction, as inferred from the contours of $F$ in Fig. 4.

The azimuthal velocity $v_{\phi}$ is symmetric with respect to $\theta=\pi/2$. The direction of $v_{\phi}>0$ means that the advection velocity in eq. (3) reduces. That is, the Hall drift-velocity is cancelled by the induced shear flow. The magnitude of $v_{\phi}$ is of the same order as $|v_{\theta}|$. Its maximum is $v_{\phi}\approx 4$ cm yr^-1 at $\theta\approx 0.2\pi,0.8\pi$, and $r\approx 0.98R$, as shown in the right panel of Fig. 4. As Lander & Gourgouliatos (2019) discussed with respect to a two-dimensional plane-parallel simulation, the plastic flow is important in the low-viscosity region, where $\nu=10^{36}$-$10^{37}$ g cm^-1 s^-1. The viscosity coefficient (eq.(27)) in our model decreases in the radial direction: $\nu=10^{37}$g cm^-1 s^-1 at $r=0.95R$ and $\nu=10^{36}$g cm^-1 s^-1 at $r=0.97R$. Thus, high-velocity regions in the vicinity of the surface correspond to low-viscosity regions. As the viscosity coefficient decreases, the spatial region relevant to $\nu<10^{37}$g cm^-1 s^-1 extends inwards, and the motion becomes global.

Figure 4 presents a snapshot obtained at $0.02\tau_{\rm H}\approx 1.3\times 10^{4}$yr, but the overall structure is almost fixed with respect to time. In addition, the spatial profile does not depend to a significant extant on the viscosity. In contrast, the magnitude of the velocity changes with the magnetic evolution and substantially depends on the viscosity. As estimated in eq.(34), the induced velocity is proportional to $\nu^{-1}$. We found that the velocity is proportional to $\nu^{-1}$ in high-viscosity models ($\gg\nu_{c}$), and that the scaling does not hold in our models with $0.2\nu_{c}$ and $\nu_{c}$. That is, the low-viscosity models with $0.2\nu_{c}$ and $\nu_{c}$ correspond to a nonlinear regime, in which the magnetic fields are also modified as a result of the back-reaction.

In Fig. 5, the magnetic fields of a high-viscosity model with $5\nu_{c}$ are presented at two specific times: $0.05\tau_{\rm H}$ and at $0.2\tau_{\rm H}$. They represent a rapidly decaying phase at the early time and a slowly decaying phase. After the time $0.2\tau_{\rm H}$, the time variation becomes much slower as inferred from the energy variation presented in Fig. 3. The azimuthal current $j_{\phi}$ indicated by the color contours in the two left panels does not change significantly in the distribution. The current is preserved near the core–crust boundary. However, the magnitude decreases roughly by half at this interval. The poloidal field lines indicated by the contours of $\Psi$ are slightly moved outwards, but the end points at the surface are almost fixed as the initial form. That is, the external dipole field is unchanged.

The evolution of toroidal magnetic fields is presented in the two right panels of Fig. 5. The field $B_{\phi}$ is set as zero as our initial condition, but it is produced by the Hall drift (see eq.(36)). The spatial pattern depends on the initial dipolar profile ($\propto\nabla_{\theta}\Psi\propto\cos\theta$) and electron fraction ($\propto\nabla_{r}\chi$). Therefore, the toroidal field is anti-symmetric with respect to $\theta=\pi/2$. The strength of the induced toroidal fields is of the order $|B_{\phi}|\sim B_{0}$, where $B_{0}$ is a magnitude of the dipolar polar cap at the surface. This value is small, as the maximum of $B_{r}$ is $\sim B_{0}$ and that of $B_{\theta}$ is $\sim 10B_{0}$ in the crust. $|B_{\phi}|\sim B_{0}$ is understood by observing eq.(36): the toroidal field $B_{\phi}$ is related to a smaller component $B_{r}(\propto\nabla_{\theta}\Psi)$ and $\nabla_{r}\chi$, which is also small owing to an almost flat electron-fraction distribution.

The results for a low-viscosity model with $0.2\nu_{c}$ are presented in Fig. 6. As shown in the two left panels, the distribution of $j_{\phi}$ is changed from the initial model. In addition to a peak caused by the initial condition near the core–crust boundary, two strong current regions appear in the vicinity of the surface with a middle latitude: $0.2\pi<\theta<0.4\pi$ and $0.4\pi<\theta<0.8\pi$. This new distribution affects the external magnetic field. We found that the field is almost described by dipole ($l=1$) and octa-pole $(l=3)$ components, although the latter is approximately one order of magnitude smaller than that of the dipole.

The toroidal magnetic field in a low-viscosity model with $0.2\nu_{c}$ is shown in the two right panels of Fig. 6. On comparing it with Fig. 5, it is clear that the field strength of $B_{\phi}$ increases on reducing the viscosity coefficient. The overall viscosity coefficient differs by a factor 25, but the maximum of $|B_{\phi}|$ in Fig. 6 increases by a factor of a few than that in Fig. 5. The magnetic-field dissipation is also enhanced, such that high-velocity fluid motion does not effectively result in a stronger toroidal field. It is also interesting to compare the low-viscosity model with the high-viscosity model in a late-time spatial pattern. In Fig. 5, we observe $B_{\phi}<0$ for the majority of the upper region, whereas $B_{\phi}>0$ for the majority of the upper region in Fig. 6. The directions of $B_{\phi}$ at $0.2\tau_{\rm H}$ in the low- and high-viscosity models are opposite.

One might be confused by the dependency of our results on the viscosity $\nu$. An inviscid fluid corresponds to $\nu=0$ and we have the Hall evolution in that case, as shown in subsection 3.3. However, we here show that a smaller value of $\nu$ effectively changes from the magnetic field evolution obtained without considering the bulk motion. In order to elucidate this fact, we first consider a model in mechanics: a particle is falling along a vertical axis under gravity and a drag force. Equation of motion is given by $dv/dt=-\nu v-g$, where $\nu$ and $g$ are positive constants. By integrating this equation, we find that the velocity $v$ approaches a constant $v_{*}\equiv-g\nu^{-1}$ for $t\gg t_{*}\equiv\nu^{-1}$. The terminal velocity $v_{*}$ satisfies $dv/dt=0$. As the drag coefficient $\nu$ decreases, $v_{*}$ increases. The timescale $t_{*}$ also increases. In an extreme case $t_{*}$ may exceed the timescale concerned and approximation of $v=v_{*}$ breaks down. Thus, we should be careful about the inviscid limit, $\nu=0$. Our quasi-stationary assumption corresponds to $dv/dt=0$(see section 2.1) and the bulk velocity is determined by the condition. In the high-viscosity model, the resultant velocity is too small to change magnetic field evolution. That is, the results are the same as those obtained without bulk velocity. As the viscosity $\nu$ decreases, the velocity increases and differences in magnetic fields become prominent. For example, toroidal field strength increases with the decrease of $\nu$. Thus, the high-viscosity model with $5\nu_{c}$ is the same as the Hall evolution, whereas the low-viscosity model with $0.2\nu_{c}$ is quite different. Like an illustrated model, it is impossible to obtain inviscid results by taking the limit of $\nu=0$, since our quasi-stationary assumption is no longer valid. The timescale $\tau_{\rm v}$ (eq.(33)) becomes too short.