256
dimensional models involving a temperature averaged along latitude circles. Expressing
time in years one may typically think of an integral mean
1 j
5
1 2 11'
u(t,ifJ) = 2071' -5 0 T(t+s,ifJ,>.)d>.ds,
in case that the seasonal variation is not accounted for, or of a mean of the form
2
5 j~ 1211'
u(t, ifJ) = 1l7r j~5 _~ 0 T(t + s + j, ifJ, >.)d>. ds,
if the seasons are to be resolved. Here, T stands for the synoptic (measured) temperature
near ground, and throughout the time unit is a year.
Of course, land-water distribution and orography are sufficient reasons to retain the
longitudinal dependence, too, as it was done in [North et al. (1983), Hetzer et al. (1989)]
and this setting will be used in the following. Thus, we suppose that the temperature u
under consideration lives on a time-space cylinder R+ x M with M a compact, connected,
oriented Riemannian manifold without boundary, e.g. the Euclidean two-sphere in the
climatological application.
The balance equation of energy states for a one-layer e~ergy balance model without
seasonal resolution that
c(x)8t u(t,x) = net radiation flux - horizontal heat flux,
where the inertia term c denotes the heat capacity. Clearly, the principal difficulty,
which arises here, is how to model the horizontal heat transport, a convective (advective
in the language of meteorologists) process on the synoptic time scale, without falling
back upon a momentum equation. Mostly (d. [Diaz (1995), Hetzer/Schmidt (1992)]
for nonlinear approaches), a linear diffusive approximation has been used, which leads
to the following reaction-diffusion equation when employing a one-layer model
(1) c(x)8t u(t,x) - div(k(·) gradu(t, ·))(x) = It Q(x)[l - O'(x,u(t,x))]- g(u(t,x))
x E M, t > O. The unknown function u is nonnegative, temperature in Kelvin, the
differential operators div and grad act on the spatial part, and they are understood
with respect to the Riemannian metric of M. The other quantities involved are the
incoming solar radiation flux Q, which is latitude-dependent, the solar constant /1-, a
dimensional models involving a temperature averaged along latitude circles. Expressing
time in years one may typically think of an integral mean
1 j
5
1 2 11'
u(t,ifJ) = 2071' -5 0 T(t+s,ifJ,>.)d>.ds,
in case that the seasonal variation is not accounted for, or of a mean of the form
2
5 j~ 1211'
u(t, ifJ) = 1l7r j~5 _~ 0 T(t + s + j, ifJ, >.)d>. ds,
if the seasons are to be resolved. Here, T stands for the synoptic (measured) temperature
near ground, and throughout the time unit is a year.
Of course, land-water distribution and orography are sufficient reasons to retain the
longitudinal dependence, too, as it was done in [North et al. (1983), Hetzer et al. (1989)]
and this setting will be used in the following. Thus, we suppose that the temperature u
under consideration lives on a time-space cylinder R+ x M with M a compact, connected,
oriented Riemannian manifold without boundary, e.g. the Euclidean two-sphere in the
climatological application.
The balance equation of energy states for a one-layer e~ergy balance model without
seasonal resolution that
c(x)8t u(t,x) = net radiation flux - horizontal heat flux,
where the inertia term c denotes the heat capacity. Clearly, the principal difficulty,
which arises here, is how to model the horizontal heat transport, a convective (advective
in the language of meteorologists) process on the synoptic time scale, without falling
back upon a momentum equation. Mostly (d. [Diaz (1995), Hetzer/Schmidt (1992)]
for nonlinear approaches), a linear diffusive approximation has been used, which leads
to the following reaction-diffusion equation when employing a one-layer model
(1) c(x)8t u(t,x) - div(k(·) gradu(t, ·))(x) = It Q(x)[l - O'(x,u(t,x))]- g(u(t,x))
x E M, t > O. The unknown function u is nonnegative, temperature in Kelvin, the
differential operators div and grad act on the spatial part, and they are understood
with respect to the Riemannian metric of M. The other quantities involved are the
incoming solar radiation flux Q, which is latitude-dependent, the solar constant /1-, a