255
ice-edge. The papers of Diaz [Diaz (1995)] and North [North (1995)] in this volume deal
with such models.
Here, I am going to focus on Sellers-type models, where the albedo depends continuously on temperature. The underlying idea is that regions permanently covered with
ice or snow have boundary zones, where one finds seasonal ice or snow cover, a fact that
allows for a gradual transition in the albedo function.
In Sections 2. - 4., I am going to describe and study what I like to call the basic
model, where we suppose a direct albedo-temperature coupling and ignore the effect
of seasonal forcing. In a one-layer setting we are led to an autonomous, but spatially
heterogeneous parameter-dependent reaction-diffusion equation on the Euclidean unitsphere. The (model) climates are identified with the stable steady states (stationary
solutions) of the reaction-diffusion system, and we are interested in the dependence of
these solutions on the solar constant.
As we will see, already such "simple" diagnostic models provide apparently unsurpassable obstactles for a sole analytical treatment, and most findings in the climatological
literature on this subject are based on computer simulations. This survey will however focus on analytical aspects, and the reader is referred to [GhiljChildress (1987)] or
[Hetzer et al. (1989)] for numerical methods and results and further literature.
Sections 5. - 10. are devoted to a model extension that includes a seasonal cycle.
Neglecting this cycle is surely questionable when addressing long-time issues such as
the ice-ages. The Milankovitch forcing could be enhanced significantly by seasonal
effects.
Finally, we account in Sections 1l. - 14. for the long reponse times of the continental
ice-sheets by giving up the direct albedo-temperature coupling. This leads to a reactiondiffusion equation involving a delay term.
2. Model Equation.
The fundamental hypothesis, which underlies all energy balance climate models, is
that it is possible to predict long term means of temperature solely from the energy
budget. Starting with the early work of Budyko [Budyko (1969)] and Sellers [Sellers
(1969)] in the late sixties, most researchers in this area have focused on so-called one-
ice-edge. The papers of Diaz [Diaz (1995)] and North [North (1995)] in this volume deal
with such models.
Here, I am going to focus on Sellers-type models, where the albedo depends continuously on temperature. The underlying idea is that regions permanently covered with
ice or snow have boundary zones, where one finds seasonal ice or snow cover, a fact that
allows for a gradual transition in the albedo function.
In Sections 2. - 4., I am going to describe and study what I like to call the basic
model, where we suppose a direct albedo-temperature coupling and ignore the effect
of seasonal forcing. In a one-layer setting we are led to an autonomous, but spatially
heterogeneous parameter-dependent reaction-diffusion equation on the Euclidean unitsphere. The (model) climates are identified with the stable steady states (stationary
solutions) of the reaction-diffusion system, and we are interested in the dependence of
these solutions on the solar constant.
As we will see, already such "simple" diagnostic models provide apparently unsurpassable obstactles for a sole analytical treatment, and most findings in the climatological
literature on this subject are based on computer simulations. This survey will however focus on analytical aspects, and the reader is referred to [GhiljChildress (1987)] or
[Hetzer et al. (1989)] for numerical methods and results and further literature.
Sections 5. - 10. are devoted to a model extension that includes a seasonal cycle.
Neglecting this cycle is surely questionable when addressing long-time issues such as
the ice-ages. The Milankovitch forcing could be enhanced significantly by seasonal
effects.
Finally, we account in Sections 1l. - 14. for the long reponse times of the continental
ice-sheets by giving up the direct albedo-temperature coupling. This leads to a reactiondiffusion equation involving a delay term.
2. Model Equation.
The fundamental hypothesis, which underlies all energy balance climate models, is
that it is possible to predict long term means of temperature solely from the energy
budget. Starting with the early work of Budyko [Budyko (1969)] and Sellers [Sellers
(1969)] in the late sixties, most researchers in this area have focused on so-called one-