123
4 Predictability of the second kind
4.1 Uncertainty in forcing
In section 2, initial value problems were referred to as predictions of the
first kind. In a prediction of the second kind, we estimate how (the attractor of) a given dynamical system responds to a change in some prescribed
parameter or variable. The response of climate to doubling CO2, or of the
stratosphere to an increase in CFCs, or of an atmospheric GCM to a prescribed change in SST, are all predictions of the second kind. Uncertainties
in such predictions may arise from the accuracy in the prescribed change
itself, or from uncertainties in model formulation. (In practice, of course,
many forecasts do not fall exclusively into either of these two categories).
Even though predictions of the second kind are, by construction, not
sensitive to initial conditions, the underlying instabilities of the flow play
an important role in determining the associated predictability. To see this,
let us apply the singular vector analysis discussed in section 2 to a forced
problem. Consider then the generalisation of (2.2) to
dx
dt = Mlx + f(t)
(4.1)
As before, we let us integrate this equation over the finite time interval
[tt, to]. Using the tangent propagator L(t, to) (cf equation 2.3), then the
solution to (4.1) can be written as
l
tl
X(tl) = L(tt, to)x(to) + L(tt, s)f(s)ds
to
(4.2)
From (4.2) it can be seen that the effect of an initial error x(to) can be
replicated by the action of the impulsive forcing f(t} = x(to)l5(t - to).
Hence, the maximum response Ilx(tl)11 from such a normalised impulsive
forcing occurs when f(t) = "1(to)l5(t - to), where "1(tO) is the dominant
singular vector at initial time associated with the interval [tb to]. More
generally, if f(t) is a normalised impulsive forcing f(t) = x(s)l5(t - s), tl <
S < to then the maximum response Ilx(tl)11 can be induced by choosing x(s)
to be the dominant initial singular vector for the interval [tl, s]. Putting
this together we can see that if f( t) is any spatially normalised forcing, then
the maximum response at iI will be obtained by setting f(t) = "[t1,tj(t), the
dominant initial singular vector over the interval [tl, t].
Now let us interpret the forcing f(t) as an uncertainty in model formulation which, for the sake of argument, we shall assume arises principally
4 Predictability of the second kind
4.1 Uncertainty in forcing
In section 2, initial value problems were referred to as predictions of the
first kind. In a prediction of the second kind, we estimate how (the attractor of) a given dynamical system responds to a change in some prescribed
parameter or variable. The response of climate to doubling CO2, or of the
stratosphere to an increase in CFCs, or of an atmospheric GCM to a prescribed change in SST, are all predictions of the second kind. Uncertainties
in such predictions may arise from the accuracy in the prescribed change
itself, or from uncertainties in model formulation. (In practice, of course,
many forecasts do not fall exclusively into either of these two categories).
Even though predictions of the second kind are, by construction, not
sensitive to initial conditions, the underlying instabilities of the flow play
an important role in determining the associated predictability. To see this,
let us apply the singular vector analysis discussed in section 2 to a forced
problem. Consider then the generalisation of (2.2) to
dx
dt = Mlx + f(t)
(4.1)
As before, we let us integrate this equation over the finite time interval
[tt, to]. Using the tangent propagator L(t, to) (cf equation 2.3), then the
solution to (4.1) can be written as
l
tl
X(tl) = L(tt, to)x(to) + L(tt, s)f(s)ds
to
(4.2)
From (4.2) it can be seen that the effect of an initial error x(to) can be
replicated by the action of the impulsive forcing f(t} = x(to)l5(t - to).
Hence, the maximum response Ilx(tl)11 from such a normalised impulsive
forcing occurs when f(t) = "1(to)l5(t - to), where "1(tO) is the dominant
singular vector at initial time associated with the interval [tb to]. More
generally, if f(t) is a normalised impulsive forcing f(t) = x(s)l5(t - s), tl <
S < to then the maximum response Ilx(tl)11 can be induced by choosing x(s)
to be the dominant initial singular vector for the interval [tl, s]. Putting
this together we can see that if f( t) is any spatially normalised forcing, then
the maximum response at iI will be obtained by setting f(t) = "[t1,tj(t), the
dominant initial singular vector over the interval [tl, t].
Now let us interpret the forcing f(t) as an uncertainty in model formulation which, for the sake of argument, we shall assume arises principally