Adaptivity with moving grids

Adaptivity with moving grids 119
and z ∈ [0,H] represents height. (A shallow atmosphere model is used with
H ≪ L, together with an f-plane approximation which uses a locally flat
approximation to the Earth’s curvature.) This model also includes both the
potential temperature θ (relative to a reference temperature θ 0 ) and Coriolis
effects. It is described in detail in Cullen (2006) and takes the form
Du
Dt − fv + P x =0,
Dv Cg
+ fu− (z − H/2) = 0,
Dt θ 0
Dθ
− Cv =0,
Dt
Dw
Dt + P z − gθ =0,
θ 0
u x + w z =0.
Here D is the advective (total) derivative, f is the Coriolis parameter (assumed
constant), g is the gravitational constant and C = −θ y is assumed
constant. All variables are periodic in x, withw and P x vanishing at z =0
and z = H. From certain initially smooth data (as described by Nakamura
(1994)), it is possible (Cullen 2006) for the solutions of the Eady problem
to develop severe fronts in a small number of days, and some sort of adaptive
mesh is needed to resolve the fine structure of the solution close to the
fronts. In Figure 5.11 we present the solution to the Eady problem close
to the formation of a severe tropical storm, obtained by using a pressurecorrection
method on a semi-staggered grid, looking at the contours in the
horizontal and vertical coordinates (x, z) of the longitudinal wind speed v
and potential temperature θ.
We consider two different monitor functions coupled to PMA to find adaptive
meshes for this problem. In the first case we take the arclength monitor
function
M 1 = √ 1+|∇v| 2 .
In the second case we take the monitor function M 2 to be an estimate of the
potential vorticity q of the solution, so that M 2 is taken to be the maximum
eigenvalue of the matrix
Q =
( )
vx + f v z
,
θ x θ z
for which q =det(Q). The resulting meshes are shown in Figure 5.12. In
both cases we see well-structured and regular meshes with good resolution
at the boundaries and of the front, and with no evidence of mesh tangling.
However, use of the potential vorticity monitor function M 2 leads to a mesh
which more precisely follows the physical solution. Further details are given
in Walsh et al. (2009).