11. SIMILARITY SCALING
In Section 10 we introduced a non-dimensional parameter called the Lundquist
number, denoted by S . This is just one of many non-dimensional parameters that can
appear in the formulations of both hydrodynamics and MHD. These generally express
the ratio of the time scale associated with some dissipative process to the time scale
associated with either wave propagation or transport by flow. These are important
because they define regions in parameter space that separate flows with different physical
characteristics. All flows that have the same non-dimensional parameters behave in the
same way. This property is called similarity scaling.
First consider viscous hydrodynamics. The equation of motion is
 V

 V  V   p   2 V .
 t


(11.1)
Here    /  is called the kinematic viscosity. We now introduce dimensionless
variables, signified by a tilde (%
) . For example, we choose to measure density in units of
3
 0 (kg/m ), i.e.,   3 means   30 (kg/m3). We measure lengths in units of L ,
velocity in units of V0 , pressure in units of p0 , and time in units of t 0 . Introducing this
ansatz into Equation (11.1), we have
% t 0V0
V
%V
%p  t 0  %
%2 V
% 
%  t 0 p0 %
% ,
(11.2)


%V

2
%
t
L
V0 0 L
L
where   L . We can choose our normalization values so that t 0V0 / L  1 ; this sets
the characteristic time scale as t 0  L / V0 . Then the coefficient of the first term on the
right hand side becomes p0 / 0V02 , which will be unity if we chose to measure the
pressure in terms of p0  0V02 (i.e., twice the characteristic kinetic energy). The
coefficient of the last term is then  / V0 L . It is customary to define the Reynolds’
number as
Re 
LV0

.
(11.3)
so that Equation (11.2) becomes
V
1
1
 V  V   p   2 V ,
t

Re
(11.4)
where we have now dropped the tilde notation and all variables are to be considered
dimensionless. In this form it is clear that the solution of Equation (11.4) depends only
on the Reynolds’ number. That means that all flows with the same Reynolds’ number
look the same when scaled to their characteristic variables. This is called similarity
scaling. The only thing that matters is the ratio LV0 /  . Consider a system with
characteristic dimension L , flow speed V0 , and kinematic viscosity  (for example, a air
with speed V0 blowing over a ship of length floating in water), and compare it to another
1
system with the same V0 (wind speed) and viscosity (air and water), but with length
l  L . The systems will not look the same, even qualitatively. This is because the
Reynolds’ number in the first case is Re  LV0 /  , but the Reynolds’ number in the
second case is Re  lV0 /   (l / L)Re  Re ; the Reynolds’ number is too small. In
order to make them appear the same using the same materials (i.e., air and water), the
wind velocity must increase by a factor of L / l . This is why many movie scenes of ships
in storms don’t look realistic; they were filmed with a model ship in a bathtub, and the
Reynolds’ number is too small! On the positive side, similarity scaling is the basis for
wind tunnel experiments, which have been tremendously important in the development of
advanced aircraft.
The kinematic viscosity  in Equation (11.1) has the dimensions of a diffusion
coefficient, m2/sec. Indeed, if we drop the advection and pressure force, the velocity is
seen to satisfy a diffusion equation
V
  2 V .
t
The characteristic time for viscous diffusion is    L2 /  , and the ratio of the viscous
diffusion time scale to the flow time scale t 0  L / V0 is   / t 0  LV0 /   Re . The
Reynolds’ number is fundamentally a ratio of the characteristic time scales of the system.
Now consider the combination of Faraday’s law and Ohm’s law (sometimes called
the induction equation)
B

   V  B    2 B .
t
0
(11.5)
Introducing non-dimensional variables and applying the same procedure as above, we
find
B
1 2
   V  B  
B ,
t
RM
(11.6)
where
RM 
LV0
( /  0 )
(11.7)
is the magnetic Reynolds’ number. The resistive diffusion time associated with Equation
(11.5) is  R  L2 / ( / 0 ) , and so  R / t 0  LV0 / ( / 0 )  RM . Again, the magnetic
Reynolds’ number is the ratio of the resistive diffusion time to the flow time.
If we choose instead V0  VA , the (as yet unmotivated) Alfvén time, then the
magnetic Reynolds’ number becomes S   R /  A , the Lundquist number that was
introduced in Section 10.
Now consider MHD, and, for simplicity of discussion, we let   0  constant . We
now must retain the Lorentz force J  B in the equation of motion. Measuring the
2
current density in units of J 0  B0 / 0 L (from  0 J    B ), and transforming to nondimensional variables, as before, we find that the coefficient of the non-dimensional
Lorentz force is
J 0 B0t 0
B2 t0
V2
 0
 A2 ,
0V0
0 0 LV0 V0
(11.8)
This strongly suggests measuring the velocity in terms of the Alfvén speed, whose square
is VA2  B02 / 0 0 . Then the pressure is measured in terms of twice the magnetic energy
density, p0  0VA2  B02 / 0 (which shows that the Alfvén speed is the speed at which
the kinetic energy equals the magnetic energy). The Reynolds’ number becomes
Se    /  A  LVA /  , which we will call the viscous Lundquist number (for lack of a
better name). With these choices, the (constant density) non-dimensional MHD
equations (neglecting energy) become
1
 V

 V  V  p  J  B  2 V ,
 t

Se

(11.9)
and
B
1
   V  B    2 B .
t
S
(11.10)
Solutions of the coupled MHD system appear the same (i.e., are similar) if both Se and S
are the same. Situations in which either Se or S (or both) are different will behave
differently.
There are several other non-dimensional parameters that appear in the literature that
are combinations of Se and S . For example, Pr  S / Se   / ( / 0 ) is called the
magnetic Prandtl number. It measures the relative effects of viscous and resistive
diffusion. Similarly, H  SSe is called the Hartmann number. It is important in
differentiating regimes in certain MHD flows, and also different operating regimes of
some present magnetic fusion experiments.
We have implied that the Reynolds’ number (and other non-dimensional parameters)
can differentiate regimes in which systems that satisfy the same equations behave quite
differently. This can be understood qualitatively as follows. Consider the case of
sheared flow. Its effect is to distort the fluid, as shown in the figure below.
The effect of diffusion is to smooth, or relax, the shear, and hence the distortion, as
shown below.
3
Both of these processes are at work simultaneously. The Reynolds’ number is the ratio of
the time scale associated with the smoothing and distortion processes, Re    / t0 . When
Re  1 the fluid is distorted faster than it can relax, and when Re  1 the fluid is
relaxed faster than it can be distorted. Smoothing and distortion occur on the same time
scale when Re ~ 1 , or on a length scale L0 ~  / V0 . Thus, flow with very large
Reynolds’ number will tend to look distorted and disorganized, and the velocity field will
look “spiky” (also called “turbulent”), while flow with very low Reynolds’ number will
be exceedingly smooth, like molasses. Flows with intermediate Reynolds’ number will
appear to be smooth, organized, and “laminar”. These flow regimes are illustrated below.
From left to right, these figures can be thought of either as representing the same scale
length with increasing viscosity, or representing the same viscosity with decreasing scale
length. In either case, the characteristic length scale on which the structure is resolved is
L0 .
Similar remarks apply to the structure of the magnetic field as a function of either the
magnetic Reynolds’ number RM , or the Lundquist number S . However, in this case the
structure in the current density is even sharper than that of the magnetic field,
since J ~ B / x . The spikes in the structure of the current density are called current
sheets. These will become of central importance when we discuss reconnection and
resistive instabilities.
4
There are other non-dimensional parameters associated with thermal conduction,
rotation, etc., all of which measure the relative importance of various physical effects.
5