Changes between Version 2 and Version 3 of DataScaling

Timestamp:

07/10/11 17:59:35 (14 years ago)

Author:

Brandon Shroyer

Comment:

—

DataScaling

@@ -7 +7 @@
 AstroBEAR handles the problem by dividing a ''scaling constant'' out of certain variables.  The equations of ideal MHD are scale-free, so the scaling of variables does not change the results of the simulation.  This does not hold true if physical source terms such as cooling or gravity are present; during source term steps, physical scaling must be reintroduced.
 
+Quantities divided by a scaling term are sometimes referred to as ''scaled'' quantities.  These scaled quantities are dimensionless, being described in terms of [ComputationalUnits computational units].
+
+[[BR]]
 == Scaling in AstroBEAR ==
 
@@ -13 +16 @@
 The most basic physical scaling options are set by the user in the [PhysicsDataExplained physics.data] file:
 
-||nScale||number density scale (cm^-3^)||
-||rScale||density scale (g cm^-3^)||
-||pScale||pressure scale (dynes cm^-2^)||
-||!TempScale||temperature scale (K)||
-||lScale||length scale (cm)||
+||  '''variable name'''  ||  '''scale name (units)'''  ||
+||{{{nScale}}}||number density scale (cm^-3^)||
+||{{{rScale}}}||density scale (g cm^-3^)||
+||{{{pScale}}}||pressure scale (dynes cm^-2^)||
+||{{{TempScale}}}||temperature scale (K)||
+||{{{lScale}}}||length scale (cm)||
 
 AstroBEAR derives other scaling constants such as magnetic pressure and velocity from these basic parameters:
-
-||!VelScale||velocity scale (cm s^-1^)||
-||BScale||magnetic field scale (esu)||
-||!TimeScale||time scale (s)||
-||!ScaleGrav*||gravity scale (cm^3^)||
-||!ScaleCool!**||cooling scale (erg^-1^ cm^3^)||
+||  '''variable name'''  ||  '''scale name (units)'''  ||  '''formula'''  ||
+||{{{VelScale}}}  ||velocity scale (cm s^-1^)||  [[latex($\sqrt{\gamma ~p_{scale} \over r_{scale}}$)]]  ||
+||{{{BScale}}}  ||magnetic field scale (esu)||  [[latex($\sqrt{p_{scale} \over {4 \pi}}$)]]  ||
+||{{{TimeScale}}}  ||time scale (s)||  [[latex($l_{scale} \over v_{scale}$)]]  ||
+||{{{ScaleGrav}}}*  ||gravity scale (cm^3^)||  [[latex($G ~r_{scale} ~{t_{scale}}^2$)]]  ||
+||{{{ScaleCool}}}!**  ||cooling scale (erg^-1^ cm^3^)||  [[latex($t_{scale} \over p_{scale}$)]]  ||
 
 *requires elliptic source terms.[[BR]]
 !**requires cooling source terms.
 
-Calculating the appropriate scales to use is not always a trivial task, especially if the important scale for your problem is a derived one.  For instance, 
+[[BR]]
+== Example of Scaling Calculations ==
+
+As an example, we will construct the scaling terms for the [wiki:AstroBearProjects/MHDClumps MHD clump] simulations.  We start with the following physical parameters for the simulation:
+
+|| ambient density || 100 cm^-3^ ||
+||ambient temperature || 10^4^ K ||
+||wind speed || Mach 25 ||
+||clump radius || 200 AU ||
+||plasma beta || 1 ||
+
+The first thing we notice is that the ambient wind speed is expressed as a [http://en.wikipedia.org/wiki/Mach_number Mach number] instead of in cm/s.  In essence, this quantity is pre-scaled to the sound speed of the problem, and does not need to be scaled any further.  Similarly, the [http://en.wikipedia.org/wiki/Beta_(plasma_physics) plasma beta] term is a dimensionless quantity that specifies the strength of the magnetic field in terms of the pressure, so there is no need to concern ourselves with the scaling of the magnetic field.
+
+The quantities that do need scaling are density, temperature, and distance.Since the clump radius is on the order of 100 AU, that seems like a reasonable value for {{{lScale}}}.  AstroBEAR assumes that distance scales are given in centimeters, and 1 AU = 1.495 x 10^13^ cm; therefore, the scaling term in {{{physics.data}}} is:
+{{{
+lScale = 1.49598d15,
+}}}
+
+This gives us a scaled clump radius of 2 computational units.
+
+Ideally, we would like to scale the ambient density to 1.  Since we are given the number density but not the mass density, the density scales in {{{physics.data}}} should be set as follows:
+{{{
+nScale = 1.d2,
+rScale = 0.d0,
+}}}
+This causes {{{rScale}}} to be derived from {{{nScale}}}, the known quantity.
+
+Similarly, we would like to scale the ambient temperature to 1.  To do this, we set the temperature and pressure scaling options in {{{physics.data}}} as follows:
+{{{
+pScale = 0.d0,
+TempScale = 1.d4,
+}}}
+With this configuration, AstroBEAR both scales the ambient temperature to 1 and derives the pressure scale from the temperature scale.  With {{{rScale}}} and {{{pScale}}}, we can now derive values for {{{VelScale}}} and {{{BScale}}} (see '[DataScaling#Data_File_Parameters 'Data File Parameters"] above).  {{{VelScale}}} can now be used in conjunction with {{{lScale}}} to derive the {{{TimeScale}}}, which in turn allows us to obtain a value for {{{ScaleCool}}}.
+
+{{{
+#!comment
+[[BR]]
+== More Complex Scaling ==
+
+The MHD clump problem is a relatively simple [DataScaling#Example_of_Scaling_Calculations example] of scaling, since the 
+}}}
+
 {{{
 #!comment