Reconnection

GEM Challenge

This 2D setup is based on Birn+ 2001, maybe with slight modifications.

The computation is carried out in a rectangular domain Lx/2xLx/2-L_x/2 \le x\le L_x/2 and Lz/2zLz/2-L_z/2\le z \le L_z/2. The system is taken to be periodic in the x direction with ideal conducting boundaries at z=±Lz/2z=\pm L_z/2. Thus the boundary conditions on the magnetic fields at the z boundaries are Bz=Bx/z=By/z=0B_z=\partial B_x/\partial z = \partial B_y/\partial z = 0 with corresponding conditions on th eelectric fields and particle or fluid quantities. (In Vlasiator I use open boundary condition for all quantities.)

The equilibrium chosen for the reconnection challenge problem is a Harris equilibrium with a floor in the density outside of the current layer. The magnetic field is given by

Bx(z)=B0tanh(z/λ) B_x(z) = B_0 \tanh(z/\lambda)

where λ\lambda is the current sheet scale size, and the density by

n(z)=n0sech2(z/λ)+n n(z) = n_0 \text{sech}^2(z/\lambda) + n_\infty

The electron and ion temperatures, TeT_e and TiT_i, are taken to be uniform in the initial state. We assume the plasma β=(pi+pe)/pB=1\beta = (p_i+p_e)/p_B=1 initially.

The normalization of the space and time scales of the system is chosen to be the ion inertial length di=c/ωpid_i=c/\omega_{pi} and the ion cyclotron frequency Ωi1\Omega_i^{-1}, where ωpi2=n0e2/ϵ0mi\omega_{pi}^2 = n_0e^2/\epsilon_0 m_i is evaluated with the density n0n_0 and the ion gyrofrequency Ωi=eB0/mi\Omega_i = eB_0/m_i is evaluated at the peak magnetic field. The velocities are then normalized to the Alfvén speed vAv_A. In the normalized units, B0=1B_0 = 1 and n0=1n_0 = 1. Specific parameters for the simulations are Lx=25.6,Lz=12.8,λ=0.5,n/n0=0.2L_x = 25.6, L_z = 12.8, \lambda=0.5, n_\infty/n_0 = 0.2, and Te/Ti=0.2T_e/T_i = 0.2. mi/me=25m_i/m_e = 25 is assumed if required.

The initial magnetic island is specified through the perturbation in the magnetic flux,

ψ(x,z)=ψ0cos(2πx/Lx)cos(πz/Lz) \psi(x,z) = \psi_0 \cos(2\pi x/L_x)\cos(\pi z/L_z)

where the magnetic perturbation is given by B=y^×ψ\mathbf{B} = \hat{y}\times\nabla\psi, or more specifically,

B1x=ψ0(πLz)cos(2πx/Lx)sin(πz/Lz)B1z=ψ0(2πLx)sin(2πx/Lx)cos(πz/Lz)\begin{aligned} B_{1x} &= -\psi_0\Big(\frac{\pi}{L_z}\Big)\cos(2\pi x/L_x)\sin(\pi z/L_z) \\ B_{1z} &= \psi_0\Big(\frac{2\pi}{L_x}\Big)\sin(2\pi x/L_x)\cos(\pi z/L_z) \end{aligned}

In normalized units ψ0=0.1\psi_0 = 0.1, which produces an initial island width which is comparable to the initial width of the current layer. The rationale for such a large initial perturbation is to put the system in the nonlinear regime of magnetic reconnection from the beginning of the simulation.

The initial setup is shown in the figure below.

Why do we need to resolve ion inertial length

Physically, electrons and ions separate at the scale of ion inertial length. Numerically, Hall term is important only when cell size is small enough to resolve the ion initial length. The reason is as follows. The Ohm's law is

E=(U+UH)×H \mathbf{E} = -(\mathbf{U}+\mathbf{U}_H)\times\mathbf{H}

Assume the typical flow velocity is Alfvén velocity: U=VA\mathbf{U} = \mathbf{V}_A. The Hall velocity is estimated as:

UH=Jne=×Bμ0nedBμ0neΔxBμ0neΔx \mathbf{U}_H = -\frac{\mathbf{J}}{ne} = -\frac{\nabla\times\mathbf{B}}{\mu_0ne} \sim -\frac{|dB|}{\mu_0ne\Delta x} \sim -\frac{|B|}{\mu_0ne\Delta x}

The approximation dBBdB\approx B is valid for relatively coarse grid size; for fine discrete cell sizes, this approximation does not hold, so magnetic field cannot cancel out in the following estimation. Let us now assume we can make this assumption. Then the ratio of Hall velocity and Alfvén velocity is:

UHVA=c/ωpiμ0Δx \frac{|\mathbf{U}_H|}{|\mathbf{V}_A|} = \frac{c/\omega_{pi}}{\mu_0 \Delta x}

Ion inertial length is also important for PIC: if particle's velocity is assumed to be Alfvén velocity, then ion inertial length is the same as ion gyroradius.

Vlasiator

Vlasiator requires SI units as inputs, so we need unit conversions. I select a reference number density scale nref=10amu/ccn_{ref} = 10\text{amu/cc} and magnetic field Bref=10nTB_{ref} = 10\text{nT}. The dimensionless scale β\beta is also used to determine the unit of temperature.

The reference ion frequency in rad/s is

ωi,ref=nrefqi2ϵ0mi \omega_{i,ref} = \sqrt{\frac{n_{ref}q_i^2}{\epsilon_0 m_i}}

and then the reference ion inertial length in m, which is also used as the length scale, is

di=cωi,ref d_i = \frac{c}{\omega_{i,ref}}

The time scale in s is the inverse of reference gyrofrequency,

tref=1Ωi=miqiBref t_{ref} = \frac{1}{\Omega_i} = \frac{m_i}{q_i B_{ref}}

Actually there is a 2π2\pi factor missing here, since we need to convert from gyrofrequency to frequency and then take the inverse for the period. However, traditionally no one did that.

The velocity scale in m/s is the Alfvén speed

vref=Brefμ0minref v_{ref} = \frac{B_{ref}}{\sqrt{\mu_0 m_i n_{ref}}}

The temperature scale can be derived from the magnetic pressure and β\beta

Tref=pBβnkB=Bref22μ0βnkB T_{ref} = \frac{p_B \beta}{n k_B} = \frac{B_{ref}^2}{2\mu_0}\frac{\beta}{n k_B}

Inserting the initially chosen values, we have a full set of conversion factors from the normalized units to SI units: nref=107m3,Bref=108T,vref=68960m/s,lref=72030m,tref=1.04sn_{ref} = 10^7\text{m}^{-3}, B_{ref} = 10^{-8}\text{T}, v_{ref} = 68960\text{m/s}, l_{ref} = 72030\text{m}, t_{ref} = 1.04\text{s}, and Tref=288188.74KT_{ref} = 288188.74\text{K}.

Since β,Ti,Te\beta, T_i, T_e are all initially uniform, we use the boundary values to determine the temperatures. Ti=B02/n0β/(1.0+Te/Ti)=0.83T_i = B_0^2/n_0*β / (1.0 + T_e / T_i) = 0.83, Te=0.2Ti=0.17T_e = 0.2T_i = 0.17. We use the thermal speed to estimate the velocity space grid. The thermal speed scale is Vth=kBTiTref/mi=140757m/sV_{th} = \sqrt{k_B T_{i}*T_{ref}/m_i} = 140757\text{m/s}. Based on experience, I set the velocity space extent to be 20 times of VthV_{th}.

Therefore, we have all the input parameters in SI units:

Lx=1.84×106mLz=9.22×105mλ=3.6×104mn0=1.0×107m3Ti=2.4×105KTe=4.8×104KB0=1.0×108T\begin{aligned} L_x &= 1.84\times 10^6\text{m} \\ L_z &= 9.22\times 10^5\text{m} \\ \lambda &= 3.6\times 10^4 \text{m}\\ n_0 &= 1.0\times 10^7 \text{m}^{-3} \\ T_i &= 2.4\times 10^5 \text{K} \\ T_e &= 4.8\times 10^4 \text{K} \\ B_0 &= 1.0\times 10^{-8} \text{T} \end{aligned}

With nx=256nx = 256 and nz=128nz = 128, the spatial cell size is Lx/nx=0.1diL_x / nx=0.1 d_i in x and Lz/nz=0.1diL_z/nz =0.1 d_i in z. With 160 velocity space cells in each dimension, the velocity cell size is 20/160=1/820/160 = 1/8 of the background thermal velocity.

Comparison

I extracted the points from Figure 1 in Birn+ 2001 and appended Vlasiator results. The normalized reconnection rates are shown below. Without the Hall term, the reconnection rate is significantly lower than others and is comparable with ideal MHD. Resolving the ion inertial length is also important to get fast reconnection rates.