  Stellar convection is customarily described by Mixing-Length Theory, which
makes use of the mixing-length scale to express the convective flux, velocity,
and temperature gradients of the convective elements and stellar medium. The
mixing-length scale is taken to be proportional to the local pressure scale
height, and the proportionality factor (the mixing-length parameter) must be
determined by comparing the stellar models to some calibrator, usually the Sun.
No strong arguments exist to suggest that the mixing-length parameter is the
same in all stars and at all evolutionary phases. The aim of this study is to
present a new theory of stellar convection that does not require the mixing
length parameter. We present a self-consistent analytical formulation of
stellar convection that determines the properties of stellar convection as a
function of the physical behaviour of the convective elements themselves and of
the surrounding medium. This new theory is formulated starting from a
conventional solution of the Navier-Stokes/Euler equations, i.e. the Bernoulli
equation for a perfect fluid, but expressed in a non-inertial reference frame
co-moving with the convective elements. In our formalism the motion of stellar
convective cells inside convectively-unstable layers is fully determined by a
new system of equations for convection in a non-local and time-dependent
formalism. We obtain an analytical, non-local, time-dependent sub-sonic
solution for the convective energy transport that does not depend on any free
parameter. The theory is suitable for the outer convective zones of solar type
stars and stars of all mass on the main sequence band. The predictions of the
new theory are compared with those from the standard mixing-length paradigm for
the most accurate calibrator, the Sun, with very satisfactory results.
