We saw in the previous introductory article that the ice-ages appear to come and go about every 120,000 years. Their shape suggests a fast rise in temperature followed by a slow decline which is indicative of some kind of cyclic behaviour to the climate. This can occur for two reasons:
- Because there is some form of driver whose effect is cyclic. Obvious ones are the yearly cycle of the earth around the sun which obviously forces the climate to follow a similar yearly cycle on top of any other changes
- Because the climate is inherently unstable.
As no appropriate external cycle with a period of around 120,000 years capable of driving the climate has been found the first can be ruled out. Therefore is it likely the climate is in some way unstable. So what makes a system oscillate and imposes this kind of cyclic behaviour? The answer comes in terms of the criteria for a system to oscillate in this manner.
The Barkhausen stability criterion that if A is the gain of the amplifying element in the circuit and β(jω) is the transfer function of the feedback path, so that βA is the loop gain around the feedback loop of the circuit, the circuit will sustain steady-state oscillations where:
- The system has feedback
- At the frequency of oscillation the feedback has no phase shift around the feedback loop or it is delayed by an integer number of cycles
- And the loop gain around the circuit is equal to unity
So the key requirement for oscillation is that there is feedback. However a gain of precisely unity is highly unlikely and so in a practical system, the output is limited. In an electronic amplifier the output is limited because the output cannot exceed either +ve or -ve power supply. This effectively reduces the gain at these points so the output is reduced (clipped) for any change of input. In this case the the circuit will sustain steady-state oscillations where:
- The system has feedback with gain >1
- At the frequency of oscillation the feedback has no phase shift or it is delayed by an integer number of cycles.
As the gain of the amplifier increases, the output tends more and more to be either high or low until it is effectively a bi-state square wave.
Such a circuit is shown to the right in fig 2.3 and the waveforms at key points are shown in fig 2.4.
In this case, the output (blue) switches between ±1.0V. The lower pair of resistors halve this value (green) so that the positive input (+) is ±0.5V. The other input to the amplifier (-) is fed by a resistor feeding a capacitor producing a near triangular wave (red).
So, what’s this got to do with climate? The key element in this oscillator is the RC network which determines the timing. The time to change from one transition voltage to the next is proportional to R x C.
So main features of a system with oscillations are:
- Positive feedback large enough so that a change in the input feeds back as a still larger change.
- An upper & lower switching threshold each of when reached cause a change in state so that the system then tends toward the other switching threshold
- A delay or other timing element such as the RC network.
Therefore, if we see self-induced cycles in the natural world, we should expect to see all these elements and physical systems reproducing such elements such as the time delay from the RC network.
Locked or triggered Oscillation
Finally I wish to cover the issue of “locking” of an oscillation of one frequency to a cycle of another.
In the above model, the two NAND gates (U1, U2) form a monostable pulse generator such that when the input signal rises above Vref at time t1, it causes the pair of NAND gates to flip. So the output of U2 goes high and it is locked in this state even though the input then drops below Vref at t2 because U1 goes low so that it doesn’t matter what the input to U3 is. However, eventually R1 discharges C1 causing the voltage across R1 to drop and the output of U1 goes low (t3). Now as soon as U3 goes low again (t4) , the cycle will repeat. However, note that now the cycle will repeat after the delay from the time constant R1.C1 and then at the next high input. In this way the exact start of cycle is locked to the input signal, but the overall time must be greater than the delay introduced by R1.C1.