Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Lectures on basic plasma physics:
magnetohydrodynamics (MHD)
Department of applied physics, Aalto University
January 19, 2016
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Ideal MHD equations
∇·B =0
∂B
∂t
∇ × B = µ0 j
∇×E =−
∂n
+ ∇ · (nV ) = 0
∂t
∂V
ρ
+ ρ (V · ∇) V = −∇p + j × B
∂t
d
p
=0
dt ργ
E+V ×B =0
(1)
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Plasma equilibrium
Layout
1 Plasma equilibrium
2 Examples of 1-dimensional equilibria
3 Example of 2-dimensional equilibria
4 Example of 3-dimensional equilibria
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Plasma equilibrium
The pressure balance equation
Start from the MHD momentum conservation equation:
ρ
dV
+ ∇ · p − j × B ≈ 0,
dt
In a equilibrium state
dV
dt
=
∂V
∂t
(2)
+ (V · ∇) V = 0.
Hence ∇p = j × B called the pressure balance equation.
From the pressure balance it is evident that in a equilibium
state:
B · ∇p = 0 indicating that the pressure is constant along the
magnetic field lines.
j · ∇p = 0 indicating that the pressure is constant along the
lines of current.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Plasma equilibrium
Equilibrium conditions
The pressure balance equation together with the Maxwell’s
equations gives us the equilibrium conditions:
∇p = j × B
∇ × B = µ0 j
(3)
∇·B =0
Eliminating the current reveals the famous equilibrium
equation
∇p =
1
(∇ × B) × B
µ0
(4)
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Plasma equilibrium
Pressure balance
A plasma equilibrium can be considered as a internal balance
between the plasma pressure and the forces from the magentic
field.
1
1
1
2
∇p =
(∇ × B) × B =
(B · ∇) B − ∇B
(5)
µ0
µ0
2
leading to
B2
∇ p+
2µ0
=
1
[(B · ∇) B]
µ0
where p is the plasma pressure and
B2
2µ0
(6)
is the magnetic pressure.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Plasma equilibrium
Equilibrium vs. Stability
Obtaining a stable plasma equilibrium is crucial for obtaining
controlled nuclear fusion on earth.
A global solution to the pressure balance equation for present
fusion experiments and future nuclear fusion power plants is
required.
In space plenty of stable plasma equilibria exist. A global
description of these stable plasma equilibria however are not
of much interest.
Instead the local pressure balance for unstable equilibria is of
more interest as disruptions often emmit radiation which is
detected by satellites/telescopes.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Examples of 1-dimensional equilibria
Layout
1 Plasma equilibrium
2 Examples of 1-dimensional equilibria
3 Example of 2-dimensional equilibria
4 Example of 3-dimensional equilibria
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Examples of 1-dimensional equilibria
The plasma pinch: basic concept
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Examples of 1-dimensional equilibria
The plasma pinch: applications
Pinches were the first device used by mankind for controlled
nuclear fusion
Other applications can be found in X-rays and neutron
generators, electromatic forming of metal, particle beams,...
Pinches also occur naturally, examples: lightning bolts,the
aurora, solar flares,.....
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Examples of 1-dimensional equilibria
Exercises
To calculate the equilibrium of a pinch use cylindrical coordinates.
Calculate the pressure balance for:
The θ-pinch with a magnetic field in the z direction.
The z-pinch with a magnetic field in the θ direction.
The screw-pinch with the magnetic field in the θ and z
direction.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
Layout
1 Plasma equilibrium
2 Examples of 1-dimensional equilibria
3 Example of 2-dimensional equilibria
4 Example of 3-dimensional equilibria
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
The tokamak
A common problem with using one-dimensional pinches for
nuclear fusion are the end losses.
A common method of beating these end losses, is to bend the
cylinder around into a torus.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
Move from cylindrical (r, θ, z) to toroidal coordinates
(R, φ, z).
Unfortunately the magnetic field is no longer symmetric in θ.
However axisymmetry in toroidal direction remains hence
∂B
∂φ = 0.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
2-D Equilibrium
Unfortunately when a cylindrical MHD equilibrium (exp.
θ-pinch) is bend into a torus it is no longer a equilibrium.
Reason nr.1: The magnetic field is no longer symmetric about
it’s central axisresulting in a outward force.
Reason nr.2: As the current is no longer axi-symmetric there
is a associated outward force.
Hence externally applied currents or magnetic fields are
necessary to obtain an equilibrium.
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
Flux functions
To calculate the required magnetic field and currents the
equilibrium conditions for a axisymmetric tokamak need to be
solved for B = B(R, Z).
At this point it is convenient to introduce the poloidal
magnetic flux function ψ.
ψ is determined by the poloidal flux lying within each
magnetic surface and is therefor constant on that surface.
After some algebra it can be proven that the pressure
p = p (ψ) and that there exist a flux function
F = F (ψ) = RBφ that is related to the poloidal current
density (part of the exersice session).
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
Magnetic field and Current
Hence, a expression for the magnetic field and current are given in
terms of F and ψ:
1
F
∇ψ × êφ + êφ
R
R
1 dF
1
µ0 j =
∇ψ × êφ − ∆? ψêφ
R dψ
R
B=
where the elliptic operator ∆? is given by
∂
1 ∂ψ
∂2ψ
∆? ψ = R
+
∂R R ∂R
∂Z 2
(7)
(8)
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
The Grad-Shavranov equation
To obtain an expression for ψ the Grad-Shavranov equation
needs to be solved numerically with F (ψ) and p(ψ) as input:
∆? ψ = −µ0 R2
dp
1 dF 2
−
,
dψ 2 dψ
(9)
Note that this is a second order partial derivative.
As both F and p(ψ) can be nonlinear, the solutions are not
guaranteed to exist nor be unique.
Finding solutions for the Grad-Shavranov equation is a field of
expertice by itself!
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 2-dimensional equilibria
Grad-Shafranov in action
Axisymmetric
tokamak equilibrium
Contours of the
poloidal flux and
vectors of the poloidal
magnetic field
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 3-dimensional equilibria
Layout
1 Plasma equilibrium
2 Examples of 1-dimensional equilibria
3 Example of 2-dimensional equilibria
4 Example of 3-dimensional equilibria
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 3-dimensional equilibria
As of 2016, there is not a coherent analytical theory for
three-dimensional equilibria.
The general approach to finding three-dimensional equilibria is
to solve the vacuum ideal MHD equations where the
curvature vector of the magnetic field:
κ = (B · ∇) B
now also will play an important role.
(10)
Lectures on basic plasma physics: magnetohydrodynamics (MHD)
Example of 3-dimensional equilibria
Numerical solutions have yielded designs for stellarators