Selective decay of magnetic helicity
- Prove that in the limit of small resistivity, magnetic helicity
\[ H=\int_{\mathcal{V}} \vec{A} \cdot \vec{B} d V \]
decays much more slowly than magnetic energy
\[ E=\int_{\mathcal{V}}|\vec{B}|^2 d V \]
Hint: use the Schwarz inequality for \(\vec{J}\) and \(\vec{B}\), i.e. \(|\vec{J} \cdot \vec{B}|^2 \leq|\vec{J}|^2|\vec{B}|^2\).
Schnack (2009)
The conservation of cross helicity
- Prove equation 2.66 of Biskamp, (without the dissipative terms):
\[ \frac{d H_c}{d t}=-\int_S \vec{F}_C \cdot d \vec{S} \]
where
\[ \vec{H}_c=\int_{\mathcal{V}} \vec{v} \cdot \vec{B} d V \]
and
\[ F_C=\vec{v} \times(\vec{v} \times \vec{B})+\vec{B}\left(\frac{\gamma p}{(\gamma-1) \rho}+\phi_g\right) . \]
\(\phi_g\) is the gravitational potential \(\left(-\nabla \phi_g\right.\) is the gravitational acceleration and \(p, \rho\) are pressure and density. \(\vec{v}, \vec{B}\) are velocity and magnetic field. \(S\) is the whole surface bounding the plasma volume.
Nonlinear term in Fourier space
Write the Nonlinear Term for the evolution of Elsasser variables, in incompressible MHD, in the Fourier representation for the components
\[ \left(\frac{\partial \vec{z}_i^{ \pm}(\vec{k})}{\partial t}\right)_{N L}=? ? \]
where
\[ \vec{z}_i^{ \pm}(\vec{k})=\frac{1}{(2 \pi)^3} \int_{\mathcal{V}} \vec{z}_i^{ \pm}(\vec{x}) e^{(-i \vec{k} \cdot \vec{x})} d V \]