Eliminating Pressure for Incompressible MHD

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In the formulation of Elsasser’s variables $\mathbf{U} = \mathbf{v}+\mathbf{b}$ and $\mathbf{W} = \mathbf{v}-\mathbf{b}$,

$$\begin{aligned} &&\partial_t \mathbf{U} = -(\mathbf{W}\cdot\nabla)\mathbf{U}-\nabla p,\\ &&\partial_t \mathbf{W} = -(\mathbf{U}\cdot\nabla)\mathbf{W}-\nabla p,\\ &&\nabla\cdot\mathbf{U} = \nabla\cdot\mathbf{W} = 0. \end{aligned}$$

For undisturbed fluid $\mathbf{U} = V_A$ and $\mathbf{W} =- V_A$. Introducing small purturbation $\mathbf{U} = V_A+\mathbf{u}$ and $\mathbf{W} = -V_A+\mathbf{w}$

$$\begin{aligned} &&\partial_t\mathbf{u}-V_A\partial_z\mathbf{u} = -(\mathbf{w}\cdot\nabla)\mathbf{u}-\nabla p,\\ &&\partial_t\mathbf{w}+V_A\partial_z\mathbf{w} = -(\mathbf{u}\cdot\nabla)\mathbf{w}-\nabla p.\\ \end{aligned}$$

Taking the divergence of above equations

$$\begin{aligned} \nabla p = \nabla\cdot[(\mathbf{w}\cdot\nabla)\mathbf{u}]. \end{aligned}$$

In the Fourier space

$$\begin{aligned} k^2 \tilde{p} = (\mathbf{k}\cdot\tilde{\mathbf{u}}(\mathbf{k}_1))(\mathbf{k}_1\cdot\tilde{\mathbf{w}}(\mathbf{k}_2))\delta(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}). \end{aligned}$$

Therefore the Fourier transformation of $\nabla p $ gives

$$\begin{aligned} \mathbf{k}p = (\hat{\mathbf{k}}\cdot\tilde{\mathbf{u}}(\mathbf{k}_1))(\mathbf{k}\cdot\tilde{\mathbf{w}}(\mathbf{k}_2))\delta(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}), \end{aligned}$$

where the incompressible condition $\mathbf{k}_2\cdot\tilde{\mathbf{w}}(\mathbf{k}_2)=0$ is used.

Taking the Fourier transform of the perturbed equations and defining $\omega_k = V_A k_z$

$$\begin{aligned} &&(\partial_t-i\omega_k)\tilde{\mathbf{u}}(\mathbf{k}) = -\frac{i}{8\pi}\int d^3 k_1 d^3 k_2\:[\tilde{\mathbf{u}}(\mathbf{k}_1)-\hat{\mathbf{k}}(\hat{\mathbf{k}}\cdot\tilde{\mathbf{u}}(\mathbf{k}_1)](\mathbf{k}\cdot\tilde{\mathbf{w}}(\mathbf{k}_2))\delta(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k})\\ &&(\partial_t-i\omega_k)\tilde{\mathbf{w}}(\mathbf{k}) = -\frac{i}{8\pi}\int d^3 k_1 d^3 k_2\:[\tilde{\mathbf{w}}(\mathbf{k}_1)-\hat{\mathbf{k}}(\hat{\mathbf{k}}\cdot\tilde{\mathbf{w}}(\mathbf{k}_1)](\mathbf{k}\cdot\tilde{\mathbf{u}}(\mathbf{k}_2))\delta(\mathbf{k}_1+\mathbf{k}_2-\mathbf{k}) \end{aligned}$$