277
11. Models With Long Response Times.
As of yet we have been dealing with the case, where the albedo at a surface point
x E 8 2 at time t ~ 0 can be calculated from the temperature u = u( t, x) predicted by the
model. We thought of u as describing the evolution of a ten-year mean of temperature,
e.g.
4 5 Jk
u(t,x)=- L U(t+s+j,x)ds,
11
1
j=-5 -8
in case that the seasonal variation is retained and U stands for the "measured" temperature. As pointed out in the introduction this approach is problematic when addressing
long-range issues such as the glacial-interglacial variations. In particular, the long response times of the huge continental icc-sheets (tens of thousands of years) should be
accounted for, since they obviously affect the albedo and even the heat storage term
via ice-formation. If we want to do so in the framework of a one-layer energy balance
model, we should introduce a long-term average, say, over the response time of such
ice-sheets and link expansion or retreat of these sheets to that average rather than to
u(t,x).
Thus, the albedo function a = a( x, y, z) would depend on three variables, the position x on the surface (land-water distribution, height above sea-level etc.), the "model"
temperature y (cloud albedo, snow-cover in winter etc.) and on a long-term averaged
temperature z (for icc-sheets and glaciers). In order to calculate the latter, one can
employ a weighted mean of u, e.g. J~T f3( s )u( t + s, x )ds, where T denotes the "memory
range" and f3 is positive on (-T,O] and J~Tf3(s)ds = l. Likewise, the inertia term c
should now depend on x and z. This leads to the following functional reaction-diffusion
equation:
(16)
c(x, .fT f3(s )u(t + s, x )ds )atu(i, x) - div (k(·) grad u(t,· ))(:r) =
= If Q(t,:r)[I- a(x,7J(t,.T),JO (J(s)v(t +s,x)ds)]- g(u(t,x)).
-T
Again, the climates are identified with the I-periodic stable solutions of (16), which
arc in a one-to-one correspondence with the fixed points of the time-I-map induced
11. Models With Long Response Times.
As of yet we have been dealing with the case, where the albedo at a surface point
x E 8 2 at time t ~ 0 can be calculated from the temperature u = u( t, x) predicted by the
model. We thought of u as describing the evolution of a ten-year mean of temperature,
e.g.
4 5 Jk
u(t,x)=- L U(t+s+j,x)ds,
11
1
j=-5 -8
in case that the seasonal variation is retained and U stands for the "measured" temperature. As pointed out in the introduction this approach is problematic when addressing
long-range issues such as the glacial-interglacial variations. In particular, the long response times of the huge continental icc-sheets (tens of thousands of years) should be
accounted for, since they obviously affect the albedo and even the heat storage term
via ice-formation. If we want to do so in the framework of a one-layer energy balance
model, we should introduce a long-term average, say, over the response time of such
ice-sheets and link expansion or retreat of these sheets to that average rather than to
u(t,x).
Thus, the albedo function a = a( x, y, z) would depend on three variables, the position x on the surface (land-water distribution, height above sea-level etc.), the "model"
temperature y (cloud albedo, snow-cover in winter etc.) and on a long-term averaged
temperature z (for icc-sheets and glaciers). In order to calculate the latter, one can
employ a weighted mean of u, e.g. J~T f3( s )u( t + s, x )ds, where T denotes the "memory
range" and f3 is positive on (-T,O] and J~Tf3(s)ds = l. Likewise, the inertia term c
should now depend on x and z. This leads to the following functional reaction-diffusion
equation:
(16)
c(x, .fT f3(s )u(t + s, x )ds )atu(i, x) - div (k(·) grad u(t,· ))(:r) =
= If Q(t,:r)[I- a(x,7J(t,.T),JO (J(s)v(t +s,x)ds)]- g(u(t,x)).
-T
Again, the climates are identified with the I-periodic stable solutions of (16), which
arc in a one-to-one correspondence with the fixed points of the time-I-map induced