# Equation for time evolution of Elsässer fields in incompressible MHD #turbulence $(\partial_t + \mathbf{z}^\mp\cdot\nabla)\mathbf{z}^\pm = -\nabla P_* + \nu_+ \nabla^2 \mathbf{z^\pm} + \nu_-\nabla^2\mathbf{z}^\mp$ where $\mathbf{z}^\pm = \mathbf{v}\pm\mathbf{b}$ $P_* = P + b^2/2$ $\nu_\pm = (\nu\pm\eta)/2$ [[Incompressibility]] conditions: $\nabla\cdot\mathbf{u}=0,\quad\nabla\cdot\mathbf{b}=0$ ## Derivation - We start with the momentum equation for ideal MHD: ![[Momentum equation for MHD#^6dfa59]] - Ignoring gravity and assuming incompressibility ($\nabla\cdot\mathbf{v} = 0$), we get $\rho\left(\partial_t + \mathbf{v}\cdot\nabla\right)\mathbf{v} = -\nabla p + \frac{1}{c}\mathbf{j}\times\mathbf{B} + \nu\nabla^2\mathbf{v}$ - The Hall term can be expressed as $\frac{1}{c}\mathbf{j}\times\mathbf{B} = -\frac{1}{8\pi}\nabla B^2 + \frac{1}{4\pi}\mathbf{B}\cdot\nabla\mathbf{B}$ - Substituting this back into our momentum equation, we have $\rho\left(\partial_t + \mathbf{v}\cdot\nabla\right)\mathbf{v} = -\nabla p - \frac{1}{8\pi}\nabla B^2 + \frac{1}{4\pi}\mathbf{B}\cdot\nabla\mathbf{B} + \nu\nabla^2\mathbf{v}$ ^222391 - Induction equation ![[Induction equation#^53e3fa]] - Expanding the first time on the RHS and assuming $\nabla\cdot\mathbf{v}=\nabla\cdot\mathbf{B} = 0$, $\nabla\times(\mathbf{v}\times\mathbf{B}) = \mathbf{v}(\cancel{\nabla\cdot\mathbf{B}}) - \mathbf{B}(\cancel{\nabla\cdot\mathbf{v}}) + (\mathbf{B}\cdot\nabla)\mathbf{v} - (\mathbf{v}\cdot\nabla)\mathbf{B}$ - The induction equation becomes $\frac{\partial\mathbf{B}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{B} = \mathbf{B}\cdot\nabla\mathbf{v} + \eta\nabla^2\mathbf{B}$ - Dividing the induction equation by $\sqrt{4\pi\rho}$ and adding this to the [[Equation for time evolution of Elsässer fields in incompressible MHD#^222391|momentum equation]], we get $ \begin{align} &\rho\partial_t\left(\mathbf{v} + \frac{\mathbf{B}}{\sqrt{4\pi\rho}}\right) + \rho\mathbf{v}\cdot\nabla\left(\mathbf{v} + \frac{\mathbf{B}}{\sqrt{4\pi\rho}}\right) \\\,=& -\nabla \left(p + \frac{B^2}{8\pi}\right) + \frac{1}{4\pi}\mathbf{B}\cdot\nabla\mathbf{B} + (\mathbf{B}\cdot\nabla)\mathbf{v} + \mu\nabla^2\mathbf{v} + \frac{1}{\sqrt{4\pi\rho}}\eta\nabla^2\mathbf{B} \end{align} $ - Defining $\mathbf{b} = \frac{\mathbf{B}}{\sqrt{4\pi\rho}}$, $ \begin{align} &\rho\partial_t\left(\mathbf{v} + \mathbf{b}\right) + \rho\mathbf{v}\cdot\nabla\left(\mathbf{v} + \mathbf{b}\right) \\\,=& -\nabla \left(p + \frac{b^2}{2}\right) + \rho\mathbf{b}\cdot\nabla\mathbf{b} + \sqrt{4\pi\rho}(\mathbf{b}\cdot\nabla)\mathbf{v} + \rho\nu\nabla^2\mathbf{v} + \eta\nabla^2\mathbf{b} \end{align} $