  In-situ observations in the Earth's and Saturn's magnetosheaths and in the
solar wind reveal the presence of Alfv\'en vortices as intermittent structures
in the range of scales from fluid lengths down to few ion lengths. The density
and the magnetic field associated with them appear to be compressible for
higher plasma betas. Until now, only incompressible Alfv\'en vortices have been
known. Motivated by space plasma observations we develop a new model of
magnetic vortices in high-beta plasmas with anisotropic temperature, possessing
compressible density and magnetic field, whose typical size ranges from fluid
to ion scales. At magneto-fluid scales we find novel non-propagating
field-aligned cylindrical monopoles and inclined propagating dipoles. Their
transverse magnetic and velocity fluctuations are aligned, but not identical,
{and they exhibit density and compressible magnetic field fluctuations $\delta
n$ and $\delta B_\Vert$ localized inside the vortex core. In the presence of
thermal anisotropy and acoustic effects, they may be correlated or
anti-correlated $\delta n/\delta B_\Vert={\rm constant}\gtrless 0$;
fluctuations whose velocity along the magnetic field is below the ion thermal
speed are always correlated.} At ion or kinetic scales (with the smallest radii
$\sim c/\omega_{pi}, \rho_{L i}$) {and in the absence of acoustic
perturbations}, only dipolar Alfv\'en vortices survive with similar properties
as those at fluid scales, except for their $\delta n/n_0$ that reaches the
level of $\delta B_\Vert/B_0$. At kinetic scales we find also pressure balanced
dipolar structures, possessing finite parallel electric field $E_\Vert$ and
purely compressional magnetic field perturbation.
