by Daniel Brouse and Sidd Mukherjee
Climate Change Overview
🌎 THE ODDS ARE CHANGING
1-in-1,000-year weather events are no longer as rare as their name suggests.
Historic rainfall. Record-breaking heat. Flash droughts. Extreme storms. These events are increasingly challenging the assumptions built into historical weather records.
Why?
Because the climate baseline is changing — and so are the odds.
A 1-in-1,000-year event is not a permanent characteristic of the weather. It is a probability calculated from a particular climate distribution. Change the climate distribution, and you change the odds.
🔥 Extreme heat intensifies.
🌧️ Heavy rainfall becomes more dangerous.
🌊 Flood risks escalate.
🌾 Drought and soil-moisture feedbacks amplify heat.
⚡ Climate impacts increasingly interact and compound.
This is more than climate change. It is climate acceleration: a changing system in which warming activates feedbacks, feedbacks amplify warming, and interconnected impacts reshape the risks we face.
The question is no longer whether historical weather patterns are changing. It is whether our infrastructure, economies, insurance systems, and communities are adapting to the new probabilities.
THE ODDS ARE CHANGING.
Our understanding of risk must change with them.
Introduction: When 1,000-Year Disasters Become Routine
The frequency, intensity, and clustering of so-called “1-in-1,000-year” disasters have changed dramatically since the 1990s. Events once considered exceptionally rare are now occurring repeatedly, sometimes within weeks or even days of one another.
But there is a fundamental mathematical misunderstanding behind the way these events are commonly described.
A 1-in-1,000-year event does not mean a disaster happens only once every millennium. It describes a 0.1% annual probability calculated from a particular historical climate distribution.
When the climate baseline changes, the probability distribution changes. When the distribution changes, the odds change.
A rainfall event that historically had a 0.1% annual probability can become a 1%, 2%, 4%, or even higher annual probability under substantially different climate conditions.
The event itself has not changed its statistical label. The climate conditions underlying that label have changed.
This is the critical distinction: A 1-in-1,000-year disaster is not inherently a 1-in-1,000-year event. It is a 1-in-1,000-year event under a particular set of historical conditions.
Human-caused climate change is fundamentally altering those conditions. Historical probability estimates are becoming increasingly disconnected from actual and future climate risks.
And this transformation is not limited to rainfall and flooding. Extreme heat is revealing another dimension of the same problem, with temperature extremes, soil moisture, ocean heat, and atmospheric feedbacks producing increasingly disruptive consequences.
The historical odds are changing because the underlying climate system is changing.
1. Massive Increase in Climate-Related Disasters
Data tracking extreme weather disasters show a substantial upward trajectory since the end of the 20th century.
According to data published by Yale Environment 360, climate-related disasters increased by 83% when comparing the 1980–1999 period with the subsequent 20 years.
Within that increase:
- Major flood events more than doubled worldwide.
- Severe storms increased by approximately 40%.
- Extreme weather disasters became increasingly frequent and costly.
This is not simply a matter of experiencing more bad weather. It reflects a changing climate system in which the conditions supporting extreme events are becoming increasingly pronounced.
2. The Clustering of 1,000-Year Disasters
One of the most striking developments is the clustering of extreme precipitation events.
In the 1990s, a 1-in-1,000-year rainfall event was an exceptionally isolated occurrence. Today, multiple events of comparable historical statistical magnitude are appearing within remarkably short periods.
The 2022 Cluster
During a single five-week period in the summer of 2022, the United States experienced five separate rainfall events described as 1-in-1,000-year events:
- St. Louis, Missouri
- Eastern Kentucky
- Southeastern Illinois
- Death Valley, California
- Dallas, Texas
The 2025 Cluster
During a single week in July 2025, four separate 1-in-1,000-year rainfall events struck:
- Texas
- North Carolina
- New Mexico
- Illinois
These clusters demonstrate why historical return-period statistics can no longer be interpreted as though extreme events occur independently against an unchanging climate baseline.
There is an important statistical distinction: multiple events described using historical return periods do not necessarily retain those same probabilities under today’s climate. Nor does the occurrence of several events by itself establish that they were statistically independent.
The essential point is that the historical probability distribution is changing, and the old return-period labels no longer reliably describe present-day risk.
3. The Sharp Rise in Extreme Daily Rainfall
The underlying driver of many catastrophic floods is the increasing intensity of individual precipitation events.
The U.S. Environmental Protection Agency (EPA) reports that the prevalence of extreme single-day precipitation events remained relatively steady from 1910 through the 1980s, but increased substantially beginning around 1990.
In fact, nine of the ten years with the highest extreme one-day precipitation in U.S. history have occurred since 1990.
This is particularly significant because extreme rainfall can overwhelm drainage systems, culverts, stormwater infrastructure, rivers, and flood-control structures in a matter of hours.
The problem is not simply that it rains more. The intensity of individual rainfall events is changing, pushing infrastructure beyond the conditions for which it was designed.
4. The Mathematics of Changing Return Periods: The Odds Are Changing
This is arguably the most important mathematical consequence of climate change.
A 1-in-1,000-year event is not a fixed characteristic of a rainfall event. It is a statistical probability derived from a historical climate distribution.
The return period describes the probability of an event occurring under specified climate conditions. It does not guarantee how frequently that event will occur as those conditions change.
The mathematics is straightforward:
| Historical Return Period | Annual Probability |
|---|---|
| 1-in-1,000 years | 0.1% |
| 1-in-500 years | 0.2% |
| 1-in-100 years | 1% |
| 1-in-50 years | 2% |
| 1-in-25 years | 4% |
| 1-in-10 years | 10% |
These probabilities are calculated from a particular statistical distribution. They are not permanent.
If warming increases atmospheric moisture, changes precipitation intensity, and alters storm behavior, the probability distribution shifts.
An event that previously had a 0.1% annual probability might now have a 4% annual probability.
That is a change from a nominal 1,000-year return period to a 25-year return period.
The storm does not need to become 40 times larger. Its probability of occurring at that intensity needs to increase 40-fold.
This is the fundamental distinction between changing weather and changing climate risk.
The Nonstationary Climate Problem
Traditional engineering frequently assumes that historical weather statistics provide a reliable foundation for predicting future extremes.
This assumption is called stationarity: the statistical characteristics of the climate are presumed to remain sufficiently consistent over time.
Climate change violates that assumption.
As temperatures rise, the atmosphere can hold approximately 7% more water vapor for every 1°C of warming. This increases the moisture available to fuel extreme precipitation, although actual rainfall also depends on atmospheric circulation, storm dynamics, and other conditions.
Consequently, historical rainfall distributions cannot automatically be treated as reliable descriptions of present or future probabilities.
A 100-year rainfall estimate calculated using data from the 20th century may no longer represent a 1% annual probability under today’s climate.
The same applies to 500-year and 1,000-year events.
A prominent study involving climate scientist Michael Mann found that an extreme New York City flood event estimated to have a 1-in-500-year probability under preindustrial conditions has become approximately a 1-in-25-year event under contemporary climate conditions.
That represents a 20-fold increase in annual probability, from approximately 0.2% to 4%.
The Engineering Consequences
Consider a drainage system designed to accommodate a 100-year rainfall event.
Under the historical statistical model, that rainfall intensity had a 1% annual probability.
But if the climate distribution shifts and the same rainfall intensity now has a 4% annual probability, the infrastructure is facing a fundamentally different risk.
The nominal 100-year event has become a 25-year event.
Over a 30-year infrastructure lifespan, the difference is substantial:
- At 1% annual probability, the cumulative chance of at least one occurrence is approximately 26%.
- At 4% annual probability, the cumulative chance rises to approximately 71%.
The structure has not changed. The rainfall threshold has not changed. The probability of exceeding that threshold has changed.
This is why historical flood maps, drainage calculations, building standards, insurance models, and infrastructure designs must be continually reassessed against changing climate conditions.
The historical return period is not necessarily the current return period.
The most important issue is not simply that extreme weather is becoming more intense. It is that the statistical probabilities used to measure, predict, insure against, and engineer for those events are changing.
A 1-in-1,000-year event can become a 1-in-100-year event. A 100-year event can become a 25-year event. A 25-year event can become a 10-year event.
These are not merely changes in terminology. They represent substantial increases in the probability of catastrophic consequences.
5. Escalating Financial and Human Consequences
As historical weather extremes become more frequent and intense, the economic consequences are accelerating.
Consider the increase in U.S. billion-dollar weather and climate disasters:
- 1980s: Approximately 3 billion-dollar disasters per year.
- 2020s: Approximately 19 billion-dollar disasters per year.
- 2023: A record 28 separate billion-dollar disasters.
That represents more than a sixfold increase in the average annual number of billion-dollar disasters between the 1980s and 2020s.
Inland flooding alone caused approximately $230 billion in U.S. damages between 1988 and 2021, according to Climate Central. More than one-third, approximately 37%, of those financial losses were attributed to warming-induced precipitation intensification.
These figures represent more than insurance claims and infrastructure repairs.
They translate into destroyed homes, lost businesses, damaged transportation networks, disrupted supply chains, rising insurance costs, displaced families, and increasing financial pressure on communities and governments.
Climate change is no longer an abstract environmental statistic. Its economic consequences are increasingly embedded in everyday life.
PART TWO: THE HEAT ACCELERATION CRISIS
The Anatomy of Third-Derivative Heat Jerk
The same calculus of third-derivative “jerk” behavior applies to extreme heat, but the consequences can be even more immediate and severe.
In thermodynamics, temperature is related to the average kinetic energy of particles. Earth’s climate system is continuously receiving energy from the Sun and releasing energy back into space. Human-caused greenhouse gas emissions disrupt this energy balance, causing the Earth system to accumulate excess energy, with more than 90% of that excess entering the oceans.
As this energy accumulates, the statistical distribution of temperatures shifts.
The extreme edge of the temperature distribution does not simply move forward by a uniform amount. Its shape, variability, and upper tail can change, producing increasingly unusual and consequential temperature extremes.
If the 1990s through the 2010s were characterized by accelerating heat extremes, the 2020s have brought particularly dramatic examples of abrupt statistical departures from historical experience.
The concern is not simply hotter summers. It is the interaction of extreme heat with soil moisture, atmospheric circulation, ocean temperatures, nighttime conditions, human physiology, and wildfire fuels.
This is where the Climate Jerk framework becomes particularly important.
6. Shattering Temperature Records by Extraordinary Margins
In a smoothly changing temperature distribution, new records might be expected to exceed previous records by relatively small amounts.
But the June 2021 Pacific Northwest heat dome demonstrated how dramatically extreme temperatures can depart from historical experience.
Lytton, British Columbia, reached 121.3°F (49.6°C) on June 29, 2021.
This did not merely break Canada’s previous national temperature record. It exceeded it by 4.6°C, approximately 8.3°F.
The previous national record was shattered in a single week.
The event produced temperature anomalies of up to 16–20°C above normal in parts of the affected region. Researchers documented catastrophic impacts, including hundreds of attributable deaths across the region, marine-life mortality, agricultural losses, and increased wildfire activity.
The event was so extreme that attribution research found it virtually impossible without human-caused climate change.
This is an extraordinary example of statistical whiplash: an event that departed dramatically from the historical distribution used to understand regional temperature extremes.
The significance is not simply that a temperature record was broken.
The magnitude of the departure from the historical record demonstrated how rapidly the extreme tail of the temperature distribution can change.
7. Flash Droughts and Soil-Moisture Feedback Loops
Heatwaves are no longer isolated meteorological events. They can trigger powerful feedback loops that intensify regional heating.
The process works as follows:
- Extreme temperatures increase evaporation and dry out vegetation and soil.
- Soil moisture declines, sometimes rapidly enough to produce a flash drought.
- With less water available for evaporation, less incoming solar energy is consumed by evaporative cooling.
- More energy goes into heating the land surface and the air above it.
- Higher temperatures accelerate drying further, reinforcing the feedback.
This is a critical transition.
Moist soil and healthy vegetation help regulate surface temperatures through evaporation and transpiration. When those natural cooling mechanisms weaken, the land becomes increasingly susceptible to extreme heating.
The result can be a rapid escalation in regional temperatures, particularly when persistent high-pressure systems and clear skies maintain strong solar heating.
This is nonlinear amplification.
The initial heatwave contributes to soil drying, and soil drying then makes the heatwave more intense.
Heat produces drought, drought removes natural cooling, and the loss of cooling produces more heat.
This is a powerful example of the feedback coupling described by the Climate Jerk framework.
8. Marine Heatwaves and the Ocean Heat Multiplier
The oceans absorb more than 90% of the excess heat accumulating in the Earth system.
For decades, this enormous heat reservoir has moderated the rate of atmospheric warming. But ocean heat accumulation is itself transforming marine environments and influencing atmospheric conditions.
Marine heatwaves in the North Atlantic, Gulf of Mexico, and other ocean regions demonstrate the growing scale of this problem.
The 2023 North Atlantic marine heatwave was particularly extraordinary.
Coastal waters around Florida also experienced exceptional marine heat, with some localized measurements approaching or exceeding 100°F (37.8°C).
These conditions have consequences beyond marine ecosystems.
Ocean temperatures influence atmospheric moisture, storm development, coastal weather, and the thermal conditions of nearby land.
The traditional cooling influence of the ocean can weaken when coastal waters become exceptionally warm. Sea breezes may provide less cooling, while warm ocean surfaces can contribute additional heat and moisture to the atmosphere.
At night, elevated land and ocean temperatures can limit the opportunity for temperatures to fall.
The ocean is no longer simply a buffer against atmospheric warming. Its accumulated heat is increasingly influencing the conditions experienced on land.
This is another example of energy redistribution and feedback coupling across the climate system.
9. The Compounding Human and Ecological Consequences of Extreme Heat
The acceleration of extreme heat is transforming weather events into increasingly complex humanitarian, economic, and ecological emergencies.
Three consequences deserve particular attention.
A. The Loss of Nighttime Recovery
Historically, nighttime cooling provided an important period of recovery from extreme daytime temperatures.
People, animals, vegetation, buildings, and electrical infrastructure could release accumulated heat as temperatures declined.
But nighttime temperatures are rising, and in many regions nighttime warming has been particularly pronounced.
When nighttime lows remain elevated, the body receives less opportunity to recover from daytime heat exposure.
The consequences include:
- Increased cumulative cardiovascular and physiological stress.
- Greater heat-related mortality.
- Reduced sleep quality and physical recovery.
- Increased demand for overnight air conditioning.
- Greater stress on electrical grids.
- Less opportunity for buildings and infrastructure to release accumulated heat.
The danger is not just an afternoon temperature record.
It is continuous thermal exposure without adequate nighttime recovery.
B. Wet-Bulb Temperature and Human Physiological Limits
Extreme heat becomes especially dangerous when high humidity limits the body’s ability to cool itself through sweating.
Wet-bulb temperature combines the effects of heat and humidity and helps describe the limits of evaporative cooling.
A wet-bulb temperature of 35°C (95°F) has traditionally been discussed as a theoretical upper limit for human survivability under sustained exposure.
At sufficiently high wet-bulb temperatures, the human body cannot effectively shed metabolic heat through evaporation. Core body temperature rises, creating potentially fatal heat stress.
However, the 35°C threshold is not a universal safety boundary.
Experimental research has demonstrated that uncompensable heat stress can occur at substantially lower wet-bulb temperatures, depending on humidity, air temperature, exposure duration, physical activity, age, and other physiological factors.
Research involving young, healthy adults found critical limits averaging approximately 30.6°C under the tested humid conditions. Other physiological research has identified even lower survival limits under certain hot, dry conditions.
The implications are profound.
A population does not need to reach the theoretical 35°C wet-bulb threshold to experience dangerous and potentially fatal heat exposure.
The combination of temperature, humidity, exposure duration, and human vulnerability determines the actual danger.
As extreme heat and humidity intensify, the margin between tolerable conditions and dangerous physiological stress becomes increasingly narrow.
C. The Wildfire Multiplier
Extreme heat accelerates the drying of vegetation, forests, grasslands, and other combustible materials.
A prolonged heatwave can rapidly reduce fuel moisture, transforming landscapes into highly combustible environments.
When combined with drought, strong winds, and ignition sources, these conditions can produce explosive wildfire behavior.
The feedbacks are particularly damaging:
- Heat dries vegetation.
- Drought reduces soil and plant moisture.
- Dry fuels become more combustible.
- Wildfires release additional carbon and damage ecosystems.
- Forest loss reduces future carbon storage.
- Post-fire erosion and flooding compound the original damage.
A single period of extreme heat can therefore contribute to a cascade of ecological and economic consequences extending well beyond the heatwave itself.
Traditional wildfire preparedness becomes increasingly difficult when the speed of fuel drying and the intensity of fire weather exceed historical expectations.
10. Heat Jerk: From Record-Breaking Temperatures to Systemic Instability
The third derivative provides a useful framework for understanding why extreme heat is becoming increasingly disruptive.
The first derivative describes how temperatures change.
The second derivative describes how the rate of temperature change accelerates.
The third derivative describes how that acceleration itself changes.
When applied to heat extremes, this framework directs attention toward more than rising average temperatures.
It asks whether the rate of change in extreme temperatures, heatwave duration, nighttime temperatures, soil moisture, and other indicators is itself changing.
The mechanisms are interconnected:
- Atmospheric warming increases the potential for extreme temperatures.
- Persistent circulation patterns can produce prolonged heat domes.
- Soil drying reduces evaporative cooling.
- Marine heatwaves alter coastal thermal conditions.
- Elevated nighttime temperatures prevent recovery.
- Heat and drought increase wildfire risk.
- Ecosystem damage weakens natural cooling and carbon-storage capacity.
These interactions can produce abrupt departures from historical conditions.
The resulting behavior is not adequately described by a single temperature trendline.
It is a coupled system in which changes in one component can reinforce changes in others.
Heat Jerk is the abrupt amplification of extreme heat behavior within an increasingly interconnected climate system.
Demonstrating actual third-derivative behavior requires measuring the third derivative of defined climate indicators over time. But the physical feedback mechanisms and unprecedented temperature departures provide a strong basis for investigating this framework.
PART THREE: THE CALCULUS OF CLIMATE DISASTERS
From Acceleration to Climate Jerk
To understand why extreme weather is changing so dramatically, we need to look beyond the frequency of individual disasters.
We need to examine how the rate of climate change itself is changing.
In physics, the first derivative of position is velocity. The second derivative is acceleration. The third derivative is called jerk.
Jerk measures the rate at which acceleration changes.
Applied to climate analysis, this provides a useful mathematical framework for distinguishing three stages of change:
| Derivative | Physics Term | Climate Application | Visual Analogy |
|---|---|---|---|
| First | Velocity | Climate conditions are changing. Temperatures and extreme-weather indicators are increasing. | A car traveling at 60 mph. |
| Second | Acceleration | The rate of change is increasing. Climate impacts are intensifying over time. | Pressing the accelerator and increasing speed from 60 to 70 to 80 mph. |
| Third | Jerk | The rate of acceleration is changing. Abrupt shifts and increasingly rapid departures from previous trends emerge. | Suddenly stomping on the accelerator, producing a sharp jolt. |
The distinction matters.
A system experiencing acceleration is already changing faster over time. A system experiencing changing acceleration introduces another dimension: the acceleration itself is no longer following a constant trajectory.
This is the distinction between climate change, climate acceleration, and Climate Jerk.
11. The 1990s Through the 2010s: Second-Derivative Acceleration
For decades, climate change could be described primarily through second-derivative acceleration.
Greenhouse gas concentrations increased, global temperatures rose, and the hydrological cycle intensified.
The consequences were increasingly apparent:
Predictable Physical Relationships
For every 1°C of atmospheric warming, the atmosphere can hold approximately 7% more water vapor. This relationship provides a physical basis for understanding the increasing potential for extreme precipitation.
Steady Compounding
The number of billion-dollar weather disasters increased substantially over successive decades. Flooding, severe storms, heatwaves, and other climate impacts became increasingly costly.
Changing Statistical Boundaries
Historical 100-year and 1,000-year rainfall thresholds began shifting as the underlying climate conditions changed.
Temperature distributions also shifted, increasing the frequency and intensity of extreme heat.
The system was gaining speed. The climate problem was accelerating.
But acceleration is not the end of the story.
12. The 2020s and Beyond: Third-Derivative Climate Jerk
The emerging concern is that climate change is no longer following a smooth, predictable acceleration curve.
We are increasingly observing abrupt changes, nonlinear responses, and extreme departures from historical patterns.
This is the behavior described by the Climate Jerk framework.
Three interconnected mechanisms help explain the transition.
1. Tipping Points and Feedback Coupling
Climate systems are interconnected. Changes in one component can trigger additional changes elsewhere.
As critical thresholds are crossed, feedback mechanisms can amplify initial disturbances.
Ocean warming, atmospheric moisture, changing circulation patterns, ice loss, cloud changes, soil moisture, and land-surface conditions interact rather than operate independently.
The result is an increasingly coupled climate system in which one change can amplify another.
2. Nonlinear Supercharging
Climate impacts do not necessarily increase in direct proportion to global temperature.
As feedbacks become activated and interconnected, the resulting response can become disproportionately large.
A modest additional temperature increase can produce substantially greater consequences when multiple climate mechanisms reinforce one another.
This is why extreme rainfall, flooding, heat, and other climate impacts must be understood within the context of an increasingly interconnected Earth system.
The trend is no longer adequately described by looking at one variable in isolation.
3. Statistical Whiplash and Infrastructure Failure
Civil engineering depends on historical observations, probability distributions, and projected future conditions.
But infrastructure designed around historical weather patterns is increasingly vulnerable to changing extremes.
A drainage system, bridge, culvert, or flood-control structure designed using a historical 100-year rainfall estimate can become inadequate as rainfall intensity and probability distributions change.
Likewise, electrical grids, cooling systems, transportation networks, agricultural systems, and emergency services are increasingly exposed to extreme heat beyond their original design assumptions.
The problem is compounded when engineering assumptions fail to account for nonlinear changes and accelerating climate impacts.
A structure built in 2020 using historical design assumptions may already be facing conditions substantially different from those anticipated when it was constructed.
This is statistical whiplash: the growing disconnect between the climate for which infrastructure was designed and the climate it must now withstand.
THE BOTTOM LINE
The Odds Are Changing Faster Than Our Infrastructure
The first derivative describes change.
The second derivative describes acceleration.
The third derivative describes changing acceleration.
Applied to climate analysis, these distinctions help explain why the increasing frequency and intensity of extreme weather events cannot be understood simply by extending historical trends.
The climate baseline is changing. Extreme rainfall events are clustering. Heat records are being shattered. Soil-moisture feedbacks are intensifying. Ocean temperatures are rising. Nighttime recovery is diminishing. Historical return periods are shifting.
Infrastructure is being tested beyond its original design assumptions. Economic losses are increasing. Human and ecological systems are facing increasingly severe thermal and hydrological stress.
And the consequences extend far beyond individual weather disasters.
A 1-in-1,000-year event is a statistical description, not a guarantee of rarity under a changing climate.
The historical odds are no longer reliable indicators of future risk.
We are moving from climate change to climate acceleration, with increasingly abrupt and nonlinear responses emerging across interconnected Earth systems.
The same underlying mathematics applies to rainfall extremes and extreme heat: when the climate distribution changes, the odds change. When feedbacks interact, the consequences can become disproportionately large.
The question is no longer whether historical weather patterns are changing.
The question is how quickly the odds will continue to change—and whether our infrastructure, economies, ecosystems, and communities can keep up.
The Central Message
Climate change is changing the baseline.
Climate acceleration is changing the rate.
Climate Jerk is the changing acceleration—and the abrupt consequences of an increasingly coupled Earth system.
