258
1. The hypotheses about the asymptotic behavior of a and g at 0 and 00 are purely
mathematical. We can always modify those functions outside the climatologically
meaningful range between, say roughly, 50 and 500 Kelvin and fulfil these requirements.
2. The reader unfamilar with the concepts of a Riemannian manifolds may think of M
as the Euclidean unit-sphere S2 := {x E R3 : Ixl = I} and div and grad expressed
in geographical polar coordinates (A, 1jJ). By appealing to those concepts from Riemannian geometry we avoid worrying about the "singularities at the poles", which
arise from fixing one coordinate system.
3. It should be pointed out that u is actually a "fictive temperature" at sea level. The
lapse rate of 0.00653 Kelvin per meter allows to account for the temperature decrease
with height, an adjustment, which is significant in particular when modeling a and g
explicitly. Actually, the above hypotheses are flexible enough to include lapse rates
that depend on position and/or temperature, which could be desirable in order to
achieve a more acurate approximation of the vertical temperature profile.
3. Mathematical Setting.
We want to choose the cone C+(M) := {w : M --4 [0,00) continuous} as our state
space, i.e. the evolution of the temperature average under consideration in time will be
described as a curve in a space of temperature distributions for the earth. To this end
one observes first that
(2)
) _ div(kgradv(t,·))(x) _
Ut V t,x
c(x)
- 0
generates an analytic solution semigroup S = S(t)1'J in C(M), which means that S(t)1'J
is the solution at time t > 0 of (2) with initial value v(O,') = 1' J E C(M). This
can be established in quite the same way as in [Amann (1983), Stuart (1974)]. Note
that we are dealing with a closed manifold without boundary, thus the domain of the
infinitesimal generator of the semi group is in fact dense in C(M). In order to obtain
local existence and uniqueness results for (I), u(to,') = 1' J E C(M), one then extends a
and g Lipschitz-continuously to M x R. That can be achieved e.g. by setting a(x, y) :=
a(x,y+) for x E M and y E Rand g(-y) = -g(y) for y E R+. Next, we define a
1. The hypotheses about the asymptotic behavior of a and g at 0 and 00 are purely
mathematical. We can always modify those functions outside the climatologically
meaningful range between, say roughly, 50 and 500 Kelvin and fulfil these requirements.
2. The reader unfamilar with the concepts of a Riemannian manifolds may think of M
as the Euclidean unit-sphere S2 := {x E R3 : Ixl = I} and div and grad expressed
in geographical polar coordinates (A, 1jJ). By appealing to those concepts from Riemannian geometry we avoid worrying about the "singularities at the poles", which
arise from fixing one coordinate system.
3. It should be pointed out that u is actually a "fictive temperature" at sea level. The
lapse rate of 0.00653 Kelvin per meter allows to account for the temperature decrease
with height, an adjustment, which is significant in particular when modeling a and g
explicitly. Actually, the above hypotheses are flexible enough to include lapse rates
that depend on position and/or temperature, which could be desirable in order to
achieve a more acurate approximation of the vertical temperature profile.
3. Mathematical Setting.
We want to choose the cone C+(M) := {w : M --4 [0,00) continuous} as our state
space, i.e. the evolution of the temperature average under consideration in time will be
described as a curve in a space of temperature distributions for the earth. To this end
one observes first that
(2)
) _ div(kgradv(t,·))(x) _
Ut V t,x
c(x)
- 0
generates an analytic solution semigroup S = S(t)1'J in C(M), which means that S(t)1'J
is the solution at time t > 0 of (2) with initial value v(O,') = 1' J E C(M). This
can be established in quite the same way as in [Amann (1983), Stuart (1974)]. Note
that we are dealing with a closed manifold without boundary, thus the domain of the
infinitesimal generator of the semi group is in fact dense in C(M). In order to obtain
local existence and uniqueness results for (I), u(to,') = 1' J E C(M), one then extends a
and g Lipschitz-continuously to M x R. That can be achieved e.g. by setting a(x, y) :=
a(x,y+) for x E M and y E Rand g(-y) = -g(y) for y E R+. Next, we define a