Problems
389
Let Xr,s be a column vector whose kth element is the coefficient of Y r•s in the
series expansion for the velocity potential at level k. Since the spherical harmonics
are eigenfunctions of the horizontal Laplacian operator on the sphere, (7.140) is
equivalent to a linear-algebraic system for X I of the form
[
I + (ßt)Zs(s + l)B] X n + 1 = f,
aZ
r,s
where the right side f does not involve the values of any unknown functions at
time (n + I)ßt . The N unknown variables in this relatively smalllinear system
can be determined by Gaussian elimination. Additional efficiency can be achieved
by exploiting the fact that the coefficient matrix is constant in time, so its "LU"
decomposition into upper and lower triangular matrices need only be computed
once .
Some of the forcing terms that are responsible for gravity-wave propagation in
the a-coordinate equations are nonlinear. In order to obtain the preceding linearalgebraic equation for
these terms have been decomposed into a linear part
and a nonlinear perturbation by splitting the total temperature into a constant horizontally uniform reference temperature T (a) and aperturbation. As discussed in
Section 7.2.3, this decomposition imposes a constraint on the stability of the semiimplicit solution that, roughly speaking, requires the speed of the fastest-moving
gravity wave supported by the actual atmospheric structure to be only modestly
faster than the speed of the fasting-moving gravity wave in the reference state.
This stability constraint is usually satisfied by choosing an isothermal profile for
the reference state, i.e., T(a) = To (Simmons et al. 1978). A typical value for To
is 300 K.
Problems
1. Consider small-amplitude shallow-water motions on a "mid-latitude
ß -plane." In the following, x and y are horizontal coordinates oriented eastwest and north-south, respectively; the Coriolis parameter is approximated
as 10 + ßy where 10 and ß are constant; g is the gravitational acceleration;
U > 0 is a constant mean flow from west to east; u' and v' are the perturbation west-to-east and south -to-north velocities; h is the perturbation
displacement of the free surface. Define the vorticity { and the divergence
au' av'
8=-+-.
ax ay
8 as
av' au'
{ = - - - ,
ax ay
Assume that the mean flow is in geostrophic balance,
g ali
U = - - - ,
10 ay
389
Let Xr,s be a column vector whose kth element is the coefficient of Y r•s in the
series expansion for the velocity potential at level k. Since the spherical harmonics
are eigenfunctions of the horizontal Laplacian operator on the sphere, (7.140) is
equivalent to a linear-algebraic system for X I of the form
[
I + (ßt)Zs(s + l)B] X n + 1 = f,
aZ
r,s
where the right side f does not involve the values of any unknown functions at
time (n + I)ßt . The N unknown variables in this relatively smalllinear system
can be determined by Gaussian elimination. Additional efficiency can be achieved
by exploiting the fact that the coefficient matrix is constant in time, so its "LU"
decomposition into upper and lower triangular matrices need only be computed
once .
Some of the forcing terms that are responsible for gravity-wave propagation in
the a-coordinate equations are nonlinear. In order to obtain the preceding linearalgebraic equation for
these terms have been decomposed into a linear part
and a nonlinear perturbation by splitting the total temperature into a constant horizontally uniform reference temperature T (a) and aperturbation. As discussed in
Section 7.2.3, this decomposition imposes a constraint on the stability of the semiimplicit solution that, roughly speaking, requires the speed of the fastest-moving
gravity wave supported by the actual atmospheric structure to be only modestly
faster than the speed of the fasting-moving gravity wave in the reference state.
This stability constraint is usually satisfied by choosing an isothermal profile for
the reference state, i.e., T(a) = To (Simmons et al. 1978). A typical value for To
is 300 K.
Problems
1. Consider small-amplitude shallow-water motions on a "mid-latitude
ß -plane." In the following, x and y are horizontal coordinates oriented eastwest and north-south, respectively; the Coriolis parameter is approximated
as 10 + ßy where 10 and ß are constant; g is the gravitational acceleration;
U > 0 is a constant mean flow from west to east; u' and v' are the perturbation west-to-east and south -to-north velocities; h is the perturbation
displacement of the free surface. Define the vorticity { and the divergence
au' av'
8=-+-.
ax ay
8 as
av' au'
{ = - - - ,
ax ay
Assume that the mean flow is in geostrophic balance,
g ali
U = - - - ,
10 ay