Up until this point in the semester, we have been discussing the relationship between forcings and their effects in natural systems. However, many responses significantly lag their associated forcing, resulting in interesting behavior of the systems. Below are some examples.

Annual and daily temperatures: the maximum solar radiation over the course of a year occurs during the summer solstice, around June 21, but the maximum monthly temperatures in Philadelphia occur during July and August. This is because the ground has a finite response time of several weeks to changes in solar radiation, and local air temperatures are strongly controlled by the ground temperature. Likewise, the minimum monthly temperatures occur in January and February, well after the minimum solar radiation at the winter soltice. The maximum daily temperature tends to occur mid-afternoon, after the maximum solar insolation at noon (or 1 pm during daylight savings time). Oceans have longer response times, so their maximum and minimum temperatures are more muted and occur even longer following maximum and minimum insolation. On a large scale, this pattern makes coastal climates less extreme than inland climates (e.g., Anchorage versus Fairbanks) and controls the daily pattern of winds.

Underground temperatures: changes in air temperature propagate into the ground at a rate that is dictated by the thermal diffusivity of the bedrock or soil. Over timescales of decades to centuries, the lag in underground temperatures can be used to reconstruct long-term climate trends at the ground surface, with temperature perturbations deep underground reflecting long-ago temperature changes at the surface. Over shorter timescales, the seasonal oscillation in temperature produces underground temperatures that also oscillate seasonally. However, the lag time in temperature changes underground causes some underground areas to be completely out of phase with the surface, feeling maximum temperatures in winter and minimum temperatures in summer. In permafrost areas, melting at the surface during the summer does not fully propagate through the frozen ground below, generating a soupy melted layer at the surface above an impermeable ice layer below.

Predator-prey models: classic ecological models of a two-part food chain, in which the population of a predator (for example, lions) is affected by the population size of its prey (for example, gazelles). If there are many gazelles, the lion population grows, but if there are many lions, the gazelle population shrinks. Because it takes time for lions to grow to hunting maturity, the populations never settle on stable levels. Instead, their populations oscillate, with many gazelles during times with few lions, and many lions during times with few gazelles.

Traffic congestion: drivers do not respond instantly to the slow-down and speed-up of cars in front of them. This response time results in traffic congestion that propagates up-traffic. Rather than entire groups of cars accelerating together as they pass a congested area, the signal that the traffic is accelerating takes a while to reach drivers that are several cars back. That delayed response produces congestion in areas that have no local reasons for slowed traffic.

General response patterns: in general, delayed responses turn a single perturbation into a delayed, lower amplitude pulse in response (e.g., the salt tracer experiment at Mill Creek) and oscillating perturbations into delayed, lower amplitude oscillations (e.g., annual temperatures).