# In MHD, why is the magnetic force aquared?

I'm trying to understand how the following equation:

$$-\nabla P + j\ \times\ B + pg = 0$$

Reduces to:

$$\frac{P_0}{L_0}=\frac{B_0^2}{\mu_0 \rho_0}$$

In MHD, one generally ignores the displacement current density such that: $$\mathbf{j} = \frac{ 1 }{ \mu_{o} } \nabla \times \mathbf{B} \tag{0}$$ then we have the following: $$\mathbf{j} \times \mathbf{B} = \frac{ 1 }{ \mu_{o} } \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} \tag{1}$$ From vector calculus identities, we can show that: $$\mathbf{j} \times \mathbf{B} = \frac{ 1 }{ \mu_{o} } \left[ \left( \mathbf{B} \cdot \nabla \right) \mathbf{B} - \frac{1}{2} \nabla \left( \mathbf{B} \cdot \mathbf{B} \right) \right] = \frac{ \left( \mathbf{B} \cdot \nabla \right) \mathbf{B} }{ \mu_{o} } - \nabla \left( \frac{ B^{2} }{ 2 \ \mu_{o} } \right) \tag{2}$$
The factor of $$L_{o}^{-1}$$ comes from the assumption that $$\nabla \ Q \propto L^{-1} \ Q$$, where $$L^{-1}$$ is some characteristic scale length and $$Q$$ is the relevant parameter (i.e., you assume all $$Q$$ are proportional to an exponential function of height/altitude). In your particular problem, it seems the notation is $$L_{o}$$, which I am guessing represents an atmospheric scale height or some reference altitude at which the mass density is $$rho_{o}$$.