### 2.3 Scaling features of the equations

The scaled Euler equations are the same as Equations (4, 5), but without the term proportional to
. The scaled variables obtained from the Euler equations are, then, the same. Thus, scaled variables
exhibit scaling similarity, and the Euler equations are said to be invariant with respect to scale
transformations. Said differently, this means that NS Equation (4) own scaling properties (Frisch, 1995),
that is, there exists a class of solutions which are invariate under scaling transformations. Introducing a
length scale , it is straightforward to verify that the scaling transformations and
( is a scaling factor and is a scaling index) leave invariate the inviscid NS
equation for any scaling exponent , providing . When the dissipative term is
taken into account, a characteristic length scale exists, say the dissipative scale . From a
phenomenological point of view, this is the length scale where dissipative effects start to be experienced
by the flow. Of course, since is in general very low, we expect that is very small.
Actually, there exists a simple relationship for the scaling of with the Reynolds number,
namely . The larger the Reynolds number, the smaller the dissipative length
scale.
As it is easily verified, ideal MHD equations display similar scaling features. Say the following scaling
transformations and ( here is a new scaling index different from ), leave the
inviscid MHD equations unchanged, providing , , and . This
means that velocity and magnetic variables have different scalings, say , only when the scaling for
the density is taken into account. In the incompressible case, we cannot distinguish between scaling laws for
velocity and magnetic variables.