The advection equation

In Cartesian coordinates the advection operator is

$${\displaystyle \mathbf {u} \cdot \nabla =u_{x}{\frac {\partial }{\partial x}}+u_{y}{\frac {\partial }{\partial y}}+u_{z}{\frac {\partial }{\partial z}}.}$$

where ${\displaystyle \mathbf {u} =(u_{x},u_{y},u_{z})}$ is the velocity field, and ${\displaystyle \nabla }$ is the del operator (note that Cartesian coordinates are used here).

The advection equation for a conserved quantity described by a scalar field ${\displaystyle \psi }$ is expressed mathematically by a continuity equation:

$${\displaystyle {\frac {\partial \psi }{\partial t}}+\nabla \cdot \left(\psi {\mathbf {u} }\right)=0}$$

where ${\displaystyle \nabla \cdot }$ is the divergence operator and again ${\displaystyle \mathbf {u} }$ is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies

$${\displaystyle \nabla \cdot {\mathbf {u} }=0.}$$

In this case, ${\displaystyle \mathbf {u} }$ is said to be solenoidal. If this is so, the above equation can be rewritten as

$${\displaystyle {\frac {\partial \psi }{\partial t}}+{\mathbf {u} }\cdot \nabla \psi =0}$$

In particular, if the flow is steady, then

$${\displaystyle {\mathbf {u} }\cdot \nabla \psi =0}$$

which shows that ${\displaystyle \psi }$ is constant along a streamline.

If a vector quantity ${\displaystyle \mathbf {a} }$ (such as a magnetic field) is being advected by the solenoidal velocity field ${\displaystyle \mathbf {u} }$, the advection equation above becomes:

$${\displaystyle {\frac {\partial {\mathbf {a} }}{\partial t}}+\left({\mathbf {u} }\cdot \nabla \right){\mathbf {a} }=0.}$$

Here, ${\displaystyle \mathbf {a} }$ is a vector field instead of the scalar field ${\displaystyle \psi }$.

Solving the equation

A simulation of the advection equation where u = (sin t, cos t) is solenoidal.

The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous "shock" solutions (which are notoriously difficult for numerical schemes to handle).

Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes

$${\displaystyle {\frac {\partial \psi }{\partial t}}+u_{x}{\frac {\partial \psi }{\partial x}}=0}$$

where ${\displaystyle \psi =\psi (x,t)}$ is the scalar field being advected and ${\displaystyle u_{x}}$ is the ${\displaystyle x}$ component of the vector ${\displaystyle \mathbf {u} =(u_{x},0,0)}$.

Treatment of the advection operator in the incompressible Navier–Stokes equations

According to Zang,^[2] numerical simulation can be aided by considering the skew-symmetric form for the advection operator.

$${\displaystyle {\tfrac {1}{2}}{\mathbf {u} }\cdot \nabla {\mathbf {u} }+{\tfrac {1}{2}}\nabla \cdot ({\mathbf {u} }{\mathbf {u} })}$$

where

$${\displaystyle \nabla \cdot ({\mathbf {u} }{\mathbf {u} })=[\nabla \cdot ({\mathbf {u} }u_{x}),\nabla \cdot ({\mathbf {u} }u_{y}),\nabla \cdot ({\mathbf {u} }u_{z})]}$$

and ${\displaystyle \mathbf {u} }$ is the same as above.

Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities.^[3]

Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems.

$${\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} ={\tfrac {1}{2}}\nabla \|\mathbf {u} \|^{2}+\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} }$$

$${\displaystyle {\tfrac {1}{2}}\mathbf {u} \cdot \nabla \mathbf {u} +{\tfrac {1}{2}}\nabla \cdot (\mathbf {u} \mathbf {u} )={\tfrac {1}{2}}\nabla \|\mathbf {u} \|^{2}+\left(\nabla \times \mathbf {u} \right)\times \mathbf {u} +{\tfrac {1}{2}}\mathbf {u} (\nabla \cdot \mathbf {u} )}$$

This form also makes visible that the skew-symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is very challenging and there is a large scientific literature about this.