Active Region Formation through the Negative Effective Magnetic Pressure Instability

Abstract

The negative effective magnetic-pressure instability operates on scales encompassing many turbulent eddies, which correspond to convection cells in the Sun. This instability is discussed here in connection with the formation of active regions near the surface layers of the Sun. This instability is related to the negative contribution of turbulence to the mean magnetic pressure that causes the formation of large-scale magnetic structures. For an isothermal layer, direct numerical simulations and mean-field simulations of this phenomenon are shown to agree in many details, for example the onset of the instability occurs at the same depth. This depth increases with increasing field strength, such that the growth rate of this instability is independent of the field strength, provided the magnetic structures are fully contained within the domain. A linear stability analysis is shown to support this finding. The instability also leads to a redistribution of turbulent intensity and gas pressure that could provide direct observational signatures.

Appendix A: Growth rate of NEMPI

In this appendix we derive the growth rate of NEMPI, neglecting for simplicity dissipation processes, using the anelastic approximation, and assuming H _ρ =constant and ∇ _y =0. However, in contrast to the simplistic derivation of Equation (8), we employ here a consistent treatment of lateral pressure variations in the equation of motion, ignoring in Equation (20) the nonlinearity,

(28)

where $p_{\mathrm{tot}}=\overline {p}+ p_{\mathrm{eff}}$ is the total pressure (the sum of the mean gas pressure [$\overline {p}$] and the effective magnetic pressure [p _eff]). We shall take into account that the mean magnetic field is independent of y, so the mean magnetic tension vanishes. Taking twice the curl of Equation (28), and noting that , we obtain

(29)

where we have used the anelastic approximation in the form [see the derivation of Equation (4)] and the fact that under the curl the gradient can be moved to $\overline {\rho }$. If we were to ignore variations of the density, the right-hand side would reduce to $\nabla^{2}_{x} p_{\mathrm{eff}}/\overline {\rho }H_{\rho}$. In the following, however, we retain such variations, which result from the fact that q _p=q _p(β) depends both on $\overline {B}$ and on $\overline {\rho }$. The right-hand side of Equation (29) can be simplified to give

$$ \hat {\mbox {\boldmath $z$}} {}\cdot \mbox {\boldmath $\nabla $} {}\times \mbox {\boldmath $\nabla $} {}\times \biggl({p_{\mathrm{tot}}\over \overline {\rho }^2} \mbox {\boldmath $\nabla $} {}\overline {\rho }\biggr)=\nabla_x \biggl[ \biggl(\nabla_z{p_{\mathrm{tot}} \over \overline {\rho }} \biggr){\nabla_x\overline {\rho }\over \overline {\rho }} - \biggl( \nabla_x{p_{\mathrm{tot}}\over \overline {\rho }} \biggr){\nabla_z \overline {\rho }\over \overline {\rho }} \biggr]. $$

(30)

We linearize, indicating small changes by δ or subscript 1, as in Section 1, and note that

$$ \delta \biggl({p_{\mathrm{tot}}\over \overline {\rho }} \biggr)={1\over2}v_{\mathrm{A}}^2 \bigl(1-q_{\mathrm{p}}-{\mathrm{d}} {}q_{\mathrm{p}}/{\mathrm{d}} {}\ln\beta^2\bigr) \biggl(2{\delta \overline {B}_y\over B_0}-{\delta \overline {\rho }\over \overline {\rho }} \biggr), $$

(31)

while

$$ \nabla_z \biggl({p_{\mathrm{tot}}\over \overline {\rho }} \biggr)={1\over2}v_{\mathrm{A}}^2 \bigl(1-q_{\mathrm{p}}-{\mathrm{d}} {}q_{\mathrm{p}}/{\mathrm{d}} {}\ln\beta^2\bigr) {1\over H_\rho}. $$

(32)

Inserting this into Equation (30), the δρ term in Equation (31) cancels the linearized form of $\nabla_{x}\overline{\rho}$ in Equation (30), and we are left with

$$ {\partial\over\partial t} \biggl(\Delta- {1\over H_\rho} \nabla_z \biggr) \delta \overline {U}_z= 2{v_{\mathrm{A}}^2 \over H_\rho} {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}\over{\mathrm{d}} {}\beta^2} { \nabla_x^2\delta \overline {B}_y\over B_0}, $$

(33)

where we have used Equation (14). Introducing a new variable $V_{z}=\sqrt{\overline {\rho }}\,\delta \overline {U}_{z}$, and rewriting Equation (33) for V _z , we obtain

$$ {\partial\over\partial t} \biggl(\Delta- {1 \over4 H_\rho^2} \biggr) V_z= 2\sqrt{\overline {\rho }} \, {v_{\mathrm{A}}^2\over H_\rho} {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}\over{\mathrm{d}} {}\beta^2} { \nabla_x^2\delta \overline {B}_y\over B_0}. $$

(34)

Using the linearized form of Equation (6), we arrive at the following equation:

$$ {\partial^2\over\partial t^2} \biggl(\Delta- {1 \over4 H_\rho^2} \biggr) V_z(t,x,z)= - {2 v_{\mathrm{A}}^2 \over H_\rho^2} {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}\over{\mathrm{d}} {}\beta^2} \nabla_x^2 V_z. $$

(35)

It follows from Equation (35) that a necessary condition for the large-scale instability is

$$ {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}\over{\mathrm{d}} {}\beta^2} < 0 . $$

(36)

For instance, in the WKB approximation when k _z   H _ρ ≫1, i.e. when the characteristic scale of the spatial variation of the perturbations of the magnetic and velocity fields are much smaller than the density height length [H _ρ ], the growth rate of the instability reads

$$ \lambda= {v_{\mathrm{A}}\over H_\rho} \, \biggl(-2 {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}\over{\mathrm{d}} {}\beta^2} \biggr)^{1/2} \, {k_x \over k}. $$

(37)

For an arbitrary vertical inhomogeneity of the density, we seek a solution of Equation (35) in the form V _z (t,x,z)=V(z)exp(λt+ik _x   x) and obtain an eigenvalue problem:

$$ \lambda_0^{-2}(z) \biggl[1+k_x^{-2} \biggl({1\over4H_\rho^2}-\nabla_z^2 \biggr) \biggr]V_z=\lambda^{-2}V_z, $$

(38)

where λ ⁻² is the eigenvalue and

$$ \lambda_0(z)= {v_{\mathrm{A}}(z) \over H_\rho} \, \biggl(-2 {{\mathrm{d}} {}\mathcal{P}_{\mathrm{eff}}(z)\over{\mathrm{d}} {}\beta^2} \biggr)^{1/2}. $$

(39)

We are interested in the fastest growing solutions, corresponding to the maximum value of λ. We find the maximum value of λ by discretizing Equation (38) on a grid, using a second-order finite-difference scheme for the derivatives, and solving the resultant eigenvalue problem numerically. In Figure 11 the resulting values of λ are compared with λ ₀(z) and the profiles of V _z (z). The values of λ turn out to be about twice as large as the actual growth rates found in the fully compressible mean-field models, where viscosity and magnetic diffusion are included. A reasonable improvement would be to subtract the dam** rate [η _t k ²] from the ideal growth rates [λ]; see Equation (9). In Figure 12 we compare λ(k _x ) with attenuated growth rates [λ(k _x )−η _t k ²] for different values of η _t, assuming that the effective wavenumber obeys $k^{2}\approx2k_{x}^{2}$, as explained in paragraph (iii) of Section 5.

Figure 11

Upper panel: comparison between the graphs of λ ₀(z) and the corresponding eigenvalues [λ] (horizontal lines segments) for B ₀/B _eq0=0.05, 0.1, and 0.2. Lower panel: eigenfunctions [V _z (z)] for the same three cases obtained by solving Equation (38).

Figure 12

Growth rates obtained by solving Equation (38) for B ₀/B _eq0=0.1 as a function of k _x for the case η _t=0 (dashed lines) compared with cases with different values of η _t (dotted lines). The solid line applies to the case η _t=η _t0.

It is customary to obtain approximate analytic solutions to Equation (38) as marginally bound states of an associated Schrödinger equation, $\Psi''-\tilde{U}(R) \, \Psi=0$, via the transformation

$$ \Psi(R)=\sqrt{R} \, V(z), \qquad R(z) ={v_{A0}^2 \over u_{\mathrm{rms}}^2 \beta_{\mathrm{p}}^2} \,e^{z/H_\rho} , $$

(40)

where $\overline {\rho }=\overline {\rho }_{0} \, e^{-z/H_{\rho}}$ is used as mean density profile, $v_{\mathrm{A}0}=B_{0}/\sqrt{\overline {\rho }_{0}}$ is the Alfvén speed based on the averaged density, and

$$ \tilde{U}(R)= {k_x^2 \over R} \, \biggl[{H_\rho^2 \over R} +a \biggl(1- {q_{\mathrm{p0}}\over(1+R)^2} \biggr) \biggr] $$

(41)

is the potential with

$$ a={u_{\mathrm{rms}}^2 \, \beta_{\mathrm{p}}^2 \over \lambda^2} $$

(42)

being a new eigenvalue. The potential [$\tilde{U}(R)$] has the following asymptotic behavior: $\tilde{U}(R\to0)=k_{\perp}^{2} H_{\rho}^{2}/ R^{2}$ and $\tilde{U}(R\to\infty)=a/R$. For the existence of the instability, the potential $\tilde{U}(R)$ should have a negative minimum. For example, for a long wavelength instability ($k_{\perp}^{2} H_{\rho}^{2} \ll1$) and when q _p0>1, the potential $\tilde{U}(R)$ has a negative minimum, and the instability can be excited.

When the potential $\tilde{U}(R)$ has a negative minimum and since $\tilde{U}(R\to0)>0$ and $\tilde{U}(R\to\infty)>0$, there are two points R ₁ and R ₂ (the so-called turning points) at which $\tilde{U}(R)=0$. We have computed $\tilde{U}(R)$ for several values of a/H _ρ ; see Figure 13, were we plot $\tanh\tilde{U}(R)$ to show more clearly the position of the turning points. Here we quote the nondimensional values of $\tilde{\lambda}\equiv\lambda H_{\rho}/c_{\mathrm{s}}$. It turns out for a≈0.241 H _ρ the turning points collapse to R ₁=R ₂≈0.735. The numerically computed maximum value of λ corresponds to a≈0.56 H _ρ , where the turning points are R ₁≈0.735 and R ₂≈2.4. Using the equations $\tilde{U}(R_{1,2})=0$ together with Equations (41) and (42), we obtain the growth rate of the instability as

$$ \lambda= {\beta_{\star}\,u_{\mathrm{rms}} \over H_\rho} \, { [R_1 R_2 (2+R_1+R_2) ]^{1/2}\over(1+R_1) (1+R_2)} , $$

(43)

where we have used $\beta_{\star}=\beta_{\mathrm{p}}\sqrt{q_{\mathrm{p0}}}$. Note that Equation (43) is consistent with the simple estimate (12). In the critical case where R=R ₁=R ₂, we find an upper bound of $\lambda\leq\sqrt{2}\beta_{\star}\,u_{\mathrm{rms}} R/H_{\rho}(1+R)^{3/2}$.

Figure 13

$\tanh\tilde{U}(R)$ for a=0.24 (dash–triple–dotted line, $\tilde{\lambda}=0.034$, R ₁=R ₂=0.735), a=0.3 [dash–dotted line, $\tilde{\lambda}=0.030$, (R ₁,R ₂)=(0.32,1.5)], a=0.4 [dashed line, $\tilde{\lambda}=0.026$, (R ₁,R ₂)=(0.19,1.9)], a=0.56 [solid line, $\tilde{\lambda}=0.022$, (R ₁,R ₂)=(0.12,2.4)], and a=1 [dotted line, $\tilde{\lambda}=0.017$, (R ₁,R ₂)=(0.06,2.8)].