2.3 Scaling features of the equations

The scaled Euler equations are the same as Equations (4

, 5

), but without the term proportional to $R -1$ . 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 $l$ , it is straightforward to verify that the scaling transformations $l-- > cl'$ and $u --> chu'$ ( $c$ is a scaling factor and $h$ is a scaling index) leave invariate the inviscid NS equation for any scaling exponent $h$ , providing $P --> c2hP '$ . When the dissipative term is taken into account, a characteristic length scale exists, say the dissipative scale $lD$ . From a phenomenological point of view, this is the length scale where dissipative effects start to be experienced by the flow. Of course, since $n$ is in general very low, we expect that $lD$ is very small. Actually, there exists a simple relationship for the scaling of $lD$ with the Reynolds number, namely $-3/4 lD ~ LRe$ . 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 $u --> chu'$ and $B --> cbB'$ ( $b$ here is a new scaling index different from $h$ ), leave the inviscid MHD equations unchanged, providing $P --> c2bP'$ , $T --> c2hT '$ , and $r --> c2(b- h)r'$ . This means that velocity and magnetic variables have different scalings, say $h /= b$ , 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.