It's easy to understand why the takeoff N1 limit decreases with an increase in the OAT in the chart above; the engines get less dense air at higher temperatures, and less dense air means less engine power capability. But why does the takeoff N1 limit start to decrease with a decrease in the OAT below 30 degrees Celsius? For your information, the chart is an airport analysis chart for Boeing 737.
What you see is called a flat rated engine. It means the maximum thrust from the engine is constant below the flat rated temperature (usually 30°C). Above that temperature, thrust will decrease due to the EGT (exhaust gas temperature) limit. In order to achieve a constant thrust at lower temperatures, the N1 needs to be decreased accordingly.
(CFM56-5A Lufthansa Training Manual, page 4)
There are basically 3 limits that the engine faces, temperature (maximum turbine entry temperature or maximum compressor exit temperature), pressure (maximum compressor exit pressure) and stress (maximum stress in the blades as a result of spool speed).
Varying the OAT for a specific engine design will hit one of these limits. When the OAT increases, the amount of fuel you can add to the system reduces as you get over-temperature, when the OAT decreases you could add more fuel, but that will increase spool speed which is also limited. From this figure you see the effects on SOT (turbine Stator Outlet Temperature, equivalent to Turbine Inlet Temperature, TIT):
But why does the takeoff N1 limit start to decrease with a decrease in the OAT below 30 degrees Celsius?
The limit on the left is the decrease in spool speed, note that the corrected spool speed is constant (N / √(T)) so for a lower temperature, the N also decreases.
It's easy to understand why the takeoff N1 limit decreases with an increase in the OAT in the chart above
Yes it is, this is limited by the turbine inlet temperature (or a maximum exhaust compressor temperature as used in the image above). The maximum turbine inlet temperature is fixed, the higher the inlet combustor temperature (exit high pressure compressor temperature), the less fuel can be added to heat to the air in the combustor, the less energy is available to expand in the turbines.
The constraining of the power is called flat rating.
This effect can be simulated with an engine model. If we would vary OAT for a certain power setting, e.g. constant (corrected) spool speed for a large bypass turbofan engine with imposed compressor exit pressure and turbine inlet temperature limit a model would look like:
After the variation of OAT you will yield the blue performance curves for a non-limited engine, and the black dashed curves for a limited engine:
Note that the compressor outlet pressure is the limiting factor in the graph on the left, but spool speed and compressor pressure are linked. The question is which of the 2 is reached first, this depends on the input and the design constraints.