  Within finite element models of fluids, vector-valued fields such as velocity
or momentum variables are commonly discretised using the Raviart-Thomas
elements. However, when using the lowest-order quadrilateral Raviart-Thomas
elements, standard finite element discretisations of the vector transport
equation typically have a low order of spatial accuracy. This paper describes
two schemes that improve the accuracy of transporting such vector-valued fields
on two-dimensional curved manifolds.
  The first scheme that is presented reconstructs the transported field in a
higher-order function space, where the transport equation is then solved. The
second scheme applies a mixed finite element formulation to the vector
transport equation, simultaneously solving for the transported field and its
vorticity. An approach to stabilising this mixed vector-vorticity formulation
is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These
schemes are then demonstrated, along with their accuracy properties, through
some numerical tests. Two new test cases are used to assess the transport of
vector-valued fields on curved manifolds, solving the vector transport equation
in isolation. The improvement of the schemes is also shown through two standard
test cases for rotating shallow-water models.

Numerical Analysis;Numerical Analysis;Fluid Dynamics