HarrisSheet

class plasmapy.plasma.equilibria1d.HarrisSheet(B0, delta, P0=<Quantity 0. Pa>)[source]

Bases: object

Define a Harris Sheet Equilibrium.

Magnetic field will be in the \(±x\) direction and the current density will be in the \(±z\) direction in a \(\hat{x} × \hat{y} = \hat{z}\) coordinate system.

Parameters:

B0 (Quantity) – Magnitude of magnetic field in the limit of \(y → ∞\) in units convertible to teslas.
delta (Quantity) – The thickness of the current sheet in units convertible to meters.
P0 (Quantity) – The plasma pressure in the limit of \(y → ∞\) in units convertible to pascals.

Notes

A current sheet is current limited to a surface.

A Harris sheet is a 1D ideal MHD equilibrium. In resistive MHD if there is any resistivity, it won’t be a true equilibrium since the resistivity will gradually smooth the profile out over time.

A Harris sheet is often used as the initial condition for simulations of magnetic reconnection.

Examples

>>> import astropy.units as u
>>> harris_sheet = HarrisSheet(delta=3 * u.m, B0=2 * u.T)
>>> harris_sheet.magnetic_field(y=5 * u.m)
<Quantity 1.8622... T>

Methods Summary

`current_density`(y)	Compute the current density.
`magnetic_field`(y)	Compute the magnetic field.
`plasma_pressure`(y)	Compute plasma pressure.

Methods Documentation

current_density( y: Quantity, ) → Quantity[source]

Compute the current density.

\[J_z(y) = - \frac{B_0}{δ μ_0) \mathrm{sech}^2 \left( \frac{y}{δ} \right)\]

Parameters:: y (Quantity) – Orthogonal distance from the current sheet center.

Examples

>>> import astropy.units as u
>>> B0 = 1 * u.T
>>> delta = 1 * u.m
>>> P0 = 0 * u.Pa
>>> hs = HarrisSheet(B0, delta, P0)
>>> y = [-2, 0, 2] * u.m
>>> hs.current_density(y)
<Quantity [ -56222.14... , -795774.71...,  -56222.14... ] A / m2>

magnetic_field( y: Quantity, ) → Quantity[source]

Compute the magnetic field.

In this equation, \(B_0\) is the asymptotic magnitude of the magnetic field for \(y → ±∞\) and \(δ\) is the thickness of the sheet.

\[B_x(y) = B_0 \tanh \left( \frac{y}{δ} \right)\]

Parameters:: y (Quantity) – Orthogonal distance from the current sheet center.

Examples

>>> import astropy.units as u
>>> B0 = 1 * u.T
>>> delta = 1 * u.m
>>> P0 = 0 * u.Pa
>>> hs = HarrisSheet(B0, delta, P0)
>>> y = [-2, 0, 2] * u.m
>>> hs.magnetic_field(y)
<Quantity [-0.9640...,  0.        ,  0.9640...] T>

plasma_pressure( y: Quantity, ) → Quantity[source]

Compute plasma pressure.

\[p(y) = \frac{B_0^2}{2 μ_0} \mathrm{sech}^2 \left( \frac{y}{δ} \right) + p_0\]

Parameters:: y (Quantity) – Orthogonal distance from the current sheet center.

Examples

>>> import astropy.units as u
>>> B0 = 1 * u.T
>>> delta = 1 * u.m
>>> P0 = 0 * u.Pa
>>> hs = HarrisSheet(B0, delta, P0)
>>> y = [-2, 0, 2] * u.m
>>> hs.plasma_pressure(y)
<Quantity [ 28111.07..., 397887.35...,  28111.07...] Pa>