= Miles–Howard theorem
{c}
A smooth inviscid stratified parallel flow with <gradient Richardson number> at least $1/4$ everywhere has no exponentially growing two-dimensional <normal modes>. Put $a=1/2$ in the <power-transformed Taylor–Goldstein energy identity> and take its <imaginary part>:
$$
c_i\int\left[|q'|^2+k^2|q|^2+\frac{N^2-(U')^2/4}{|U-c|^2}|q|^2\right]dz=0.
$$
The integral is positive for a nonzero mode when the numerator is nonnegative, forcing $c_i=0$. This is a modal stability theorem, not a prohibition on <transient growth>. https://www.imi.kyushu-u.ac.jp/wp-content/uploads/2022/07/Maslowe.pdf[Maslowe's review] discusses the theorem and the role of critical layers.
Back to article page
