Mathematical interpretation and graphics

Gradient and Nonlinear Comparison of the Rate of Change

The formula describing the rate of change caused by human activity relative to natural dynamics is based on the function:

z = f (x, y, t)

Thus, the gradient is:

∇f = (∂f/∂x , ∂f/∂y , ∂f/∂t)

However, climate change does not progress uniformly, but includes nonlinear couplings:

∇f_I = (∂f_I/∂x_I , ∂f_I/∂y_I , ∂f_I/∂t + αf_I²)

∇f_L = (∂f_L/∂x_L , ∂f_L/∂y_L , ∂f_L/∂t + βf_L)

Where:

• ᴴ = human-induced rate of change
• ᴺ = natural dynamics
• α and β = nonlinearity coefficients, representing, for example, feedback mechanisms

Comparison ratio in the nonlinear case:

∇f_I/∇f_L >> 1, kun αf_I² >> βf_L

Dynamic Equilibrium and Nonlinear Couplings

Natural dynamics attempt to slow down the rate of change caused by human activity, but if the system reaches a critical threshold, climate change can accelerate exponentially. For example, the albedo effect of ice cover can be described with a nonlinear differential equation:

dA/dt = - κA + cAⁿ

Where:

• A = area covered by ice
• κ = constant representing the melting rate
• c, n = parameters describing feedback

If n > 1, ice melting accelerates exponentially until a tipping point is reached.

 

Mathematical Modeling of Natural and Human Influence

The nonlinear description of rate of change is based on an extended version of the Pythagorean theorem:

ds² = dx² + dy² + γ (d_x x d_y)^p

dΦ_s = dΦ_x + dΦ_y + δ (dΦ_x x dΦ_y)^q

Where γ, δ, p, and q represent nonlinear interactions.

More generally, the components of the rate of change may behave chaotically:

dΦ = ∑_n (∂Φ/∂x_n + λΦ^m)

Where λ and m influence the nonlinear dynamics of the rate of change.

Empirical Data and Nonlinearity Across Timescales

Historical data supports nonlinear effects in climate change:

• Atmospheric CO₂ concentration has increased not linearly but exponentially: from 280 ppm in pre-industrial times to over 420 ppm by 2023 (NOAA, 2023).
• Ice melting has accelerated: Greenland is now losing approximately 270 billion tons of ice annually.
• Warming is not steady, but includes rapid transitions and potential threshold phenomena.

Timescale-based analysis shows exponential trends:

• Decades: Warming has accelerated over the past 50 years, and CO₂ emissions have not stabilized.
• Centuries: At the current emission rate, warming could exceed 3°C by 2100, potentially triggering irreversible changes.
• Geological timescale: Post-Ice Age warming (4–5°C over ~10,000 years) is far slower than the current rate.

Conclusions

• The rate of change caused by human activity is not only greater than that of natural forces, but may also contain exponential and chaotic components.
• Natural dynamics attempt to respond to changes, but when a critical point is reached, balance may collapse rapidly.
• Mathematically, climate change can be described using nonlinear gradients and differential equations that include feedback.
• Empirical data confirms that mitigating climate change is urgent, as the system may contain tipping points beyond which the trajectory becomes uncontrollable.

In Closing

This model can be expanded by adding numerical simulations to study the long-term effects of nonlinear processes. In particular, more detailed analysis of critical thresholds and chaotic behavior could help us understand how quickly the impacts of climate change may escalate—and which measures would be most effective in mitigating them.

(Appendix on Temperature Distribution)

The key question is whether the number of sub-zero days has decreased or whether the entire temperature scale has shifted. These equations describe the nonlinearity of the rate of change and feedback mechanisms, which may affect the behavior of the temperature distribution. This can be analyzed, for example, as follows:

1. Shift of the entire distribution

If the entire temperature distribution shifts upward, it means that both the coldest and hottest days become warmer.
This can be represented by: T′ = T + ΔT, where ΔT is the average temperature increase.

2. Changes in extreme temperatures

If the distribution widens, the number of extremely hot days may increase, but the number of cold days may not decrease proportionally.
This can be modeled by adding a nonlinear term to the distribution’s spread:

P(T) = P₀ e ^{- (T - T} _m)²/2σ² + λTⁿ

Where σ describes the spread, and λTⁿ captures nonlinear effects such as feedback mechanisms.

3. Use of observational data

Historical data shows that the number of extremely cold days has decreased more than the number of hot days has increased. This suggests that the entire scale has shifted upward, and that the occurrence of extremes has also changed.

Summary with Example Variables and Parameters

Climate models often use the Navier–Stokes equations and involve great detail and many variables. Since 2014, I’ve been developing this model. I now pose the question: Could this model be refined to generally examine human influences to which nature attempts to respond?

In other words, can we truly represent this comparison of rates of change with rate-of-change equations—i.e., partial differentials—to determine how fast human impact is relative to natural forces? Yes, we can! I believe this is a very interesting and important approach.

When we examine the climate as a system reacting to different drivers—such as natural cycles and human-induced changes—comparing rates of change provides an excellent way to understand dynamics at a macro level. We can approach this formally with rate-of-change equations. Suppose we have a climate parameter, such as temperature T, which changes over time. Then its total rate of change can be split into two components:

∂T/∂t = (∂T/∂t) NATURE  + (∂T/∂t) HUMAN

Where:

• (∂T/∂t)_NATURE describes slowly changing natural drivers (e.g. Milankovitch cycles = ice ages / interglacials / insolation, volcanoes, solar activity)
• (∂T/∂t)_HUMAN describes faster human-induced changes (e.g., CO₂ emissions, deforestation)

From this, we can build a general model by comparing the magnitude of these rates of change:

R = |(∂T/∂t) HUMAN| / |(∂T/∂t) NATURE|

Here, R is the rate-of-change ratio. If R >> 1, human influence dominates. If R << 1, natural processes set the pace. I consider this a simple but powerful metric.

This approach can also be extended to multiple variables, such as:

• Greenhouse gases C(t)
• Ice coverage I(t)
• Ocean temperature T_OCEAN(t)
• Biodiversity B(t)

And we can write corresponding partial differential equations for each:

∂X/∂t = (∂X/∂t) NATURE + (∂X/∂t HUMAN

This kind of system allows for the estimation of response times, resilience, and equilibrium points—without needing to solve a complex system like Navier–Stokes.

1. General Model: Rate-of-Change Distribution

Let X(t) be any climate state variable—e.g., global temperature, ice mass, CO₂ concentration, biodiversity, etc. We can describe the total change as:

∂X/∂t = f NATURAL (t) + f HUMAN(t)

Where:

• f_NATURE(t): natural, slow, cyclical change
• f_HUMAN(t): human-induced, rapid, exponential or abrupt change

2. Relative Impact: Rate-of-Change Ratio

We define the ratio:

R_X (t) = |f HUMAN (t) / f NATURE (t)|

If R_X(t) > 1, human influence exceeds natural variation. If R_X(t) < 1, natural processes dominate.

3. Example Variables

Variable X(t)      Physical Meaning

T(t)                        Temperature

C(t)                        CO₂ concentration

I(t)                         Ice coverage

B(t)                       Biodiversity

O(t)                       Ocean oxygen levels

 

Each can be written in the same form:

∂X/∂t = f_{LUINTO, X} (t) + f_{IHMINEN, X} (t)

4. Parameterized Model (e.g., for CO₂)

∂C/∂t = α x cos (ωt) + b x e^kt

• α · cos(ωt) = natural variations (e.g., seasonal and cyclical changes like Milankovitch cycles = ice ages / interglacials / insolation, volcanoes, solar activity)
• b · e^(kt) = anthropogenic emissions (exponential growth)

5. Simulation Development (Python code)

(See the graphic below)

Top Graph (Muutosnopeudet ajassa = "Rates of Change Over Time")

• Luonto (hidas) → Nature (slow) (green line)

• Ihminen (nopea) → Human (fast) (red line)

• Yhteisvaikutus → Interaction (blue dashed line)

• Muutosnopeus dx/dt → Rate of Change dx/dt

Bottom Graph (ihmisen vaikutuksen suhteellinen voimakkuus) = "Relative Strenght of Human Impact")

• Suhde R(t) = |Ihminen / Luonto| → Ratio R(t) = |Human / Nature| (purple line)

• R = 1 (tasapiste) → R = 1 (equilibrium) (gray dashed line)

• Suhdeluku R(t) → Ratio R(t)

• Aika (vuotta) → Time (years)