Titolo della tesi: Geophysical Fluid Dynamics: 2D Navier-Stokes Equations on Rotational Frame, Well-Posedness and Enstrophy Cascade.
Geophysical fluid dynamics refers to the fluid dynamics of naturally occurring
flows, such as oceans and planetary atmospheres on Earth and other planets.
These flows are primarily characterized by two elements: stratification and ro-
tation. Moreover, as a first approximation, we can idealize these flows as 2D
flows due to their large horizontal extent compared with their depth.
In this thesis, we investigate the effects of rotation on the dynamics, by
neglecting stratification, in a 2D model where we incorporate the effects of the
planetary rotation by adopting the β-plane approximation, which is a simple
device used to represent the latitudinal variation in the vertical component of
the Coriolis force.
We consider the well-known 2D β-plane Navier-Stokes equations (2DβNS)
initially in the homogeneous case and then in the statistically forced case.
Our first problem focuses on the properties of well-posedness and regularity
of the solution. We derive the vorticity equation from the 2DβNS and, after
separating the mean flow from the fluctuations, we assume that zonally averaged
flow varies on scales much larger than perturbation. Under this regime, we study
well-posedness on periodic domain, leading us to determine the Reynolds stress,
which characterizes turbulent flows.
Our second problem addresses energy-related phenomena associated with
the solution of the equations. To maintain the fluid in a turbulent state, we
introduce energy into the system through a stochastic force. In the 2D case, a
scaling analysis argument indicates a direct cascade of enstrophy and an inverse
cascade of energy. We compare the behaviour of the direct enstrophy cascade
with the 2D model lacking the Coriolis force, observing that at small scales, the
enstrophy flux from larger to smaller scales remains unaffected by the planetary
rotation, confirming experimental and numerical observations. In fact, this is
the first mathematically rigorous study of the above equations. In particular,
we provide sufficient conditions to prove that at small scales, in the presence
of the Coriolis force, the so-called third-order structure function’s asymptotics
follows the third-order universal law of 2D turbulence without the Coriolis force.
We also proved again well-posedness and certain regularity properties necessary
to obtain the mentioned results since we are no longer in the deterministic case
as in the first problem.