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Normalization of MHD equations

In the set of equations that we presented in the former section, all the parameters and variables bear their original values and are not normalized. Some examples of typical parameters and variables are given below(in SI units):

Interplanetary magnetic field at 1AU........ \(7\times 10^{-9}\) Tesla

Solar wind speed at 1AU..................... \(4.5\times 10^{5}\) m/s

\(\epsilon_0\).......................................... \(8.85\times 10^{-12}\) F/m

Plasma temperature at 1AU................... \(1.0\times 10^{5}\)oK

From the above values, we can see that there are very large quantitative differences among these values. A result when we use these numbers directly in numerical calculations is that large errors may occur. One of the most common result is that the error of the result is usually unacceptably large, when one very large number is subtracted from a similarly large number of nearly identical value.

A simple way for us to over come this numerical difficulty is to normalize these equations. Normalization is the transformation of the parameters and variables of the simulation to proper units, which makes these parameters and variables not vary far away from unity. This will avoid the difficulty we meet when no normalization is performed.

We select eq.(13) as a simple example to show how the normalization is performed. For convenience, we write eq.(13) in the following form:

\begin{displaymath}\frac{\partial\rho}{\partial t}=-\nabla\cdot(\rho\textbf{v})
\end{displaymath} (22)

Generally, we use the typical values of the simulation system to perform normalization. Usually we choose local Alfven speed as the velocity unit. The typical solar wind Alfven speed at 1AU is \(V_A=4\times 10^4\)m/s, then v'=v/VA. The typical density is \(\rho_0=7\times 10^{-6}m^3\), then \(\rho'=\rho/\rho_0\). we select typical length of the simulation as \(L_0=1.0\times 10^6m\), then we have x'=x/L0, y'=y/L0, z'=z/L0. As to time, we select the typical time as T0, then t'=t/T0. Since we have set the other normalization units for other parameters, we can not choose T0 freely. In fact, to keep the original form of the equations, we need to consider the choice of the typical units to make them consistent and optimized for the simulation.

Now, we can put those normalized value into eq.(22), we can get

\begin{displaymath}\frac{\partial(\rho'\rho_0)}{\partial (t'T_0)}=
-\frac{1}{L_0}\nabla '\cdot\{(\rho'\rho_0)(\textbf{v'}V_A)\}
\end{displaymath} (23)

Further we can write eq.(23) as

\begin{displaymath}\frac{\partial\rho'}{\partial t'}=
-\frac{V_A T_0}{L_0}\nabla '\cdot(\rho ' v')
\end{displaymath} (24)

To keep the original form of eq.(45), we should set:

\begin{displaymath}\frac{V_A T_0}{L_0}=1
\end{displaymath} (25)

from which we can get \(T_0=\frac{L_0}{V_A}\).

The normalization of the whole set of MHD equations are more complex than this example, and we also need to consider the relationship between several variables to get the best typical value for normalization.

Generally, after normalization, the forms of the equations do not change. All the variables are changed to normalized variables. All the parameters, such as \(\mu_0, \epsilon_0\), are eliminated in the normalization. So the equations after normalization will be simpler.

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Next: Equations Up: Components of MHD Numerical Previous: Components of MHD Numerical