Beschreibung:
<jats:p>This paper investigates the three-dimensional stability of a Lamb–Chaplygin columnar
vertical vortex pair as a function of the vertical wavenumber <jats:italic>k</jats:italic><jats:sub><jats:italic>z</jats:italic></jats:sub>, horizontal
Froude number <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub>, Reynolds number <jats:italic>Re</jats:italic> and Schmidt number <jats:italic>Sc</jats:italic>. The horizontal
Froude number <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> (<jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> = <jats:italic>U</jats:italic>/<jats:italic>NR</jats:italic>,
where <jats:italic>U</jats:italic> is the dipole travelling velocity, <jats:italic>R</jats:italic> the dipole radius and <jats:italic>N</jats:italic> the
Brunt–Väisälä frequency) is varied in the range [0.033, ∞[
and three set of Reynolds-Schmidt numbers are investigated:
{<jats:italic>Re</jats:italic> = 10 000, <jats:italic>Sc</jats:italic> = 1}, <jats:italic>Re</jats:italic> = 1000, <jats:italic>Sc</jats:italic> = 1}, {<jats:italic>Re</jats:italic> = 200, <jats:italic>Sc</jats:italic> = 637}.
In the whole range of <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> and <jats:italic>Re</jats:italic>, the dominant mode is always antisymmetric with respect to the middle plane between the
vortices but its physical nature and properties change when <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> is varied. An elliptic
instability prevails for <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> > 0.25, independently of the Reynolds number. It manifests
itself by the bending of each vortex core in the opposite direction to the vortex periphery.
The growth rate of the elliptic instability is reduced by stratification effects but
its spatial structure is almost unaffected. In the range 0.2 < <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> < 0.25, a continuous
transition occurs from the elliptic instability to a different instability called zigzag
instability. The transitional range <jats:italic>F</jats:italic><jats:sub><jats:italic>hc</jats:italic></jats:sub> = 0.2–0.25 is in good agreement with the value
<jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> = 0.22 at which the elliptic instability of an infinite uniform vortex is suppressed by
the stratification. The zigzag instability dominates for <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> [les ] 0.2 and corresponds to a
vertically modulated bending and twisting of the whole vortex pair. The experimental
evidence for this zigzag instability in a strongly stratified fluid reported in the first
part of this study (Billant & Chomaz 2000<jats:italic>a</jats:italic>) are therefore confirmed and extended.
The numerically calculated wavelength and growth rate for low Reynolds number
compare well with experimental measurements.</jats:p><jats:p>The present numerical stability analysis fully agrees with the inviscid asymptotic
analysis carried out in the second part of this investigation (Billant & Chomaz 2000<jats:italic>b</jats:italic>)
for small Froude number <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> and long wavelength. This confirms that the zigzag
instability is related to the breaking of translational and rotational invariances. As
predicted, the growth rate of the zigzag instability is observed to be self-similar with
respect to the variable <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub><jats:italic>k</jats:italic><jats:sub><jats:italic>z</jats:italic></jats:sub>, implying that the maximum growth rate is independent
of <jats:italic>F</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub> while the most amplified dimensional wavenumber varies with <jats:italic>N</jats:italic>/<jats:italic>U</jats:italic>. The
numerically computed eigenmode and dispersion relation are in striking agreement
with the analytical results.</jats:p>