Unsteady turbulence in stably and unstably stratified flow with system rotation around
the vertical axis is analysed using the rapid distortion theory (RDT). Complete linear
solutions for the spectra, variances and covariances are obtained analytically, and
their characteristics, including the short- and long-time asymptotics and the effect
of initial conditions, are examined in detail. It has been found that the rotation
modifies the energy partition among the three kinetic energy components and the
potential energy, and the ratio of the Coriolis parameter f to the Brunt–Väisälä
frequency N, i.e. f/N, determines the final steady values of these components. The
ratio also determines the phase of the energy/flux oscillation. Depending on whether
f/N > 1 or f/N < 1, there is a phase shift of
±π/4. However, unsteady aspects are
largely dominated by stratification. This occurs because the effects of the Coriolis
parameter f appear only in the form of fk3, which vanishes for the horizontal
wavenumber components (k3 = 0), which contribute most to the energies and the
fluxes. For example, the oscillation frequency of the energies and the fluxes asymptotes
to 2N over a long time, in agreement with the stratified non-rotating turbulence. The
initial time development is also dominanted by the stratification, and the short-time asymptotics (Nt, ft
[Lt ]
1) agree with those for non-rotating stratified fluids in
the lowest-order approximation. In the special case of f = N, all the wavenumber
components oscillate in phase, leading to no inviscid decay of oscillation. This is in
contrast to the general case of f ≠ N, in which inviscid decay has been observed. For
pure rotation (f ≠ 0, N = 0), analytical solutions showed that any turbulence that
is initially axisymmetric around the rotation axis recovers exact three-dimensional
isotropy in the kinetic energy components. Comparison with previous DNS and
experiments shows that many of the unsteady aspects of the kinetic and potential
energies and the vertical density flux can be explained by the linear processes described
by RDT. Even the time development of the vertical vorticity, which would represent
the small-scale characteristics of turbulence, agrees well with DNS. For unstably
stratified turbulence, the initial processes observed in DNS and experiments, such
as the initial decay of the kinetic energy due to viscosity and the subsequent rapid
growth of the vertical kinetic energy compared to the horizontal kinetic energy, could
be explained by RDT.