A study of the distribution of energy among the different scales is performed for several systems in fluid mechanics, including the Navier–Stokes, magnetohydrodynamics and active scalars equations. It is found that all these systems possess a common structure which enables us to deduce how the energy introduced by the forcing is transferred to the scales present in the flow. It is also shown that in special cases an energy cascade will occur. The limits of this method are also considered.

1.
C. Foias, O. Manley, R. Rosa, and R. Temam, “Navier-Stokes equations and turbulence,” in Encyclopedia of Mathematics and its Applications, Vol. 83 (Cambridge University Press, Cambridge, 2001).
2.
U. Frisch, Turbulence (Cambridge University Press, Cambridge, 1995).
3.
C.
Foias
,
O.
Manley
,
R.
Rosa
, and
R.
Temam
,
C. R. Acad. Sci. Paris, Sér. I
333
,
499
(
2001
).
4.
R. Temam, Infinite-Dimensional Dynamical systems in Mechanics and Physics (Springer, New York, 1988).
5.
M.
Sermange
and
R.
Temam
,
Commun. Pure Appl. Math.
36
,
635
(
1983
).
6.
D. Biskamp, Nonlinear Magnetohydrodynamics (Cambridge University Press, Cambridge, 1993).
7.
P.
Constantin
,
SIAM Rev.
36
,
73
(
1994
).
8.
P.
Constantin
,
Commun. Math. Phys.
129
,
241
(
1990
).
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