Nonlinear Analysis: Human Activity vs. Natural Climate Dynamics

 

H. A. Sinivirta, 30 September 2025

 

Abstract

Climate change arises from the combined influence of natural variability and human activity. In this work, I present a theoretical framework that assesses the relative strength of human influence using a dimensionless rate-of-change ratio. I derive the ratio through dimensional analysis and demonstrate its applicability to CO₂ concentration, global temperature, and the size of the cryosphere. The results show that anthropogenic drivers exceed natural variability by orders of magnitude in several key variables. The framework offers a simple yet informative tool for climate-risk assessment and can serve as a bridge between detailed models and decision-making.

 

Keywords: climate change, nonlinear dynamics, dimensional analysis, anthropogenic forcing, rate-of-change ratio.

 

1. Introduction

The climate system combines slow natural processes (orbital cycles, volcanic eruptions, variations in solar activity) with accelerating human-induced drivers (greenhouse gas emissions, land-use changes, aerosol emissions). Global Climate Models (GCMs) describe these processes accurately, but their complexity makes them heavy and less suited for rapid assessment.

 

In this work, I propose a macro-level mathematical model with the following aims:

 

• to compare the relative contributions of natural and human-induced rates of change,
• to identify the significance of feedbacks and potential critical points,
• to provide a simple, dimensionless metric for climate-risk assessment.

 

2. Theoretical Framework

2.1 Notation

Define the climate parameter vector:

X = (X₁, X₂, … , Xₙ)

Each parameter Xᵢ may represent, for example, temperature, CO₂ concentration, or ice extent. Its total change can be written as:

dXᵢ/dt = F_{N, i} + F_{A, i}

Where F_{N, i} is the natural forcing and F_{A, i} the anthropogenic forcing.

 

2.2 Dimensional Analysis and Scaling

For each variable, choose a characteristic scale X_{0, i} and a time scale τ. The dimensionless variables are:

 

X̅_i  = X_i  / X_{0, i} , … t̅ = t / τ

 

Define the dimensionless rate-of-change ratio:

 

R_{X, i}  = F_{A ,i} / F_{N , i}

 

Interpretation of the ratio:

• R_X,i ≪ 1: Natural forcing dominates
• R_X,i ≈ 1: Dynamics are sensitive to disturbances
• R_X,i ≫ 1: Anthropogenic forcing dominates the behavior of the variable

 

2.3 Nonlinear Couplings

 

Natural feedback mechanisms can be written as:

 

F_{N, i} = fi (X_i)

 

where fᵢ is often linear or weakly nonlinear on short time scales (e.g., radiative equilibrium response).

 

Anthropogenic forcing is written as:

 

F_{A, i} = g_i (t)

 

Where g_i (t) is primarily an external, time-dependent forcing (e.g., emission trends). Nonlinearity becomes particularly evident in feedback couplings such as water-vapor and albedo responses, which can be

 

incorporated as dependencies:

 

F_{A, i} (f_b) = k_i X_i^mi, m_i > 1

 

This cleanly separates the baseline human forcing from state-dependent feedbacks.

 

2.4 Orbital Forcing

 

Milanković cycles form a slow background variation: M(t). On current time scales (decades to a century), orbital forcing is weak and even slightly cooling. For this reason, it can in many applications be treated as a background term:

F_{N, i} tot = f_i (X_i ) + M(t)

 

3. Discussion

 

The proposed Rₓ ratio makes the asymmetry in climate rates of change explicit: human activity affects the dynamics of variables far more rapidly than natural forcings. The ratio is analogous to the Damköhler number and can help:

 

• assess the sensitivity of the climate system,
• identify the strength of feedback mechanisms,
• prioritize emission-reduction actions.

 

Limitations:

 

• Parameter uncertainties and feedback calibration affect the results.
• The model does not replace GCMs for regional-scale assessments.
• The ratio does not capture all dynamic couplings, but serves as a macro-level indicator.

 

 

4. Conclusions

 

1. Anthropogenic rates of change exceed natural variability by several orders of magnitude.
2. Dynamic feedback mechanisms amplify human influence.
3. The Rₓ ratio provides a clear and intuitive metric for climate risks.
4. The proposed framework is suitable for both theoretical analysis and decision support.

 

Human activity is no longer merely an additional factor in the climate system — it is its central driving force.