Accurate prediction of the future state of the atmosphere is important
throughout society, ranging from the weather forecast in a few days time
to modelling the effects of a changing climate over decades and generations.
The equations which govern how the atmosphere evolves have long been
known; these are the Navier-Stokes equations, the laws of thermodynamics
and the equation of state. Unfortunately the nonlinearity of the equations
prohibits analytic solutions, so simplified models of particular
flow phenomena have historically been, and continue to be, used alongside numerical
models of the full equations.
In this thesis, the two-dimensional Eady model of shear-driven frontogenesis
(the creation of atmospheric fronts) was used to investigate how errors
made in a localised region can affect the global solution. Atmospheric fronts
are the boundary of two different air masses, typically characterised by a
sharp change in air temperature and wind direction. This occurs across
a small length of O(10 km), whereas the extent of the front itself can be
O(1000 km). Fronts are a prominent feature of mid-latitude weather systems
and, despite their narrow width, are part of the large-scale, global
solution. Any errors made locally in the treatment of fronts will therefore
affect the global solution.
This thesis uses the convergence of the Euler equations to the semigeostrophic
equations, a simplified model which is representative of the
large-scale flow, including fronts. The Euler equations were solved numerically
using current operational techniques. It was shown that highly predictable
solutions could be obtained, and the theoretical convergence rate
maintained, even with the presence of near-discontinuous solutions given by
intense fronts.
Numerical solutions with successively increased resolution showed that
the potential vorticity, which is a fundamental quantity in determining the
large-scale, balanced flow, approached the semigeostrophic limit solution.
Regions of negative potential vorticity, indicative of local areas of instability,
were reduced at high resolution. In all cases, the width of the front reduced
to the grid-scale.
While qualitative features of the limit solution were reproduced, a stark
contrast in amplitude was found. The results of this thesis were approximately
half in amplitude of the limit solution. Some attempts were made at
increasing the intensity of the front through spatial- and temporal-averaging.
A scheme was proposed that conserves the potential vorticity within the
Eady model.Open Acces
