Editing Advection (section)

===The advection equation===

The advection equation for a conserved quantity described by a [[scalar field]] <math>\psi(t,x,y,z)</math> is expressed by a [[Continuity_equation#Differential_form|continuity equation]]:
<math display="block"> \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\mathbf u}\right) =0, </math>
where [[vector field]] <math>\mathbf{u} = (u_x, u_y, u_z)</math> is the [[flow velocity]] and <math>\nabla</math> is the [[del]] operator.<ref group=note>The subscripts denote the coordinates of the vector field; not to be confused with the [[Notation_for_differentiation#Partial_derivatives|notation for partial derivatives]].</ref>  If the flow is assumed to be [[incompressible flow|incompressible]] then <math>\mathbf{u}</math> is [[solenoidal]], that is, the [[Del#Divergence|divergence]] is zero:<math display="block">\nabla\cdot{\mathbf u} = 0,</math>and (by using a [[Del#Divergence|product rule associated with the divergence]]) the above equation reduces to

<math display="block"> \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi =0.</math>

In particular, if the flow is [[steady state|steady]], then{{sfn|LeVeque|2002|p=391}}<math display="block">{\mathbf u}\cdot\nabla \psi = 0,</math>which shows that <math>\psi</math> is constant (because <math display="inline">\nabla \psi = 0</math> for any vector <math display="inline">\mathbf u</math>) along a [[Streamlines, streaklines and pathlines|streamline]]. 

If a vector quantity <math>\mathbf{a}</math> (such as a [[magnetic field]]) is being advected by the [[solenoidal]] [[velocity field]] <math>\mathbf{u}</math>, then the advection equation above becomes:

<math display="block"> \frac{\partial{\mathbf a}}{\partial t} + \left( {\mathbf u} \cdot \nabla \right) {\mathbf a} =0. </math>

Here, <math>\mathbf{a}</math> is a [[vector field]] instead of the [[scalar field]] <math>\psi</math>.