---
title: "Climate Energy Balance Interactive Lab | Sandbox Physics"
description: "Why does Earth keep warming after the heating stops increasing? A steady extra heat input keeps adding energy until warming increases heat loss enough to balance it. The deep ocean absorbs some of that energy, so the surface warms quickly at first and then approaches its final temperature much more slowly."
canonical_url: "https://sandboxphysics.com/labs/climate-energy-balance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/climate-energy-balance/"
last_modified: "2026-10-24"
---

D-X3 · The ocean slows the warming

# Climate Energy Balance

**Why does Earth keep warming after the heating stops increasing?** A steady extra heat input keeps adding energy until warming increases heat loss enough to balance it. The deep ocean absorbs some of that energy, so the surface warms quickly at first and then approaches its final temperature much more slowly.

Add a steady heat input and watch the surface warm ahead of the deep ocean. Change the heat exchange to discover why the final temperature takes much longer to arrive.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

This synthetic, global two-box model follows temperature changes from an initially balanced state after a constant step in radiative forcing. A positive restoring coefficient means warming increases energy loss to space; the coefficient includes all feedbacks as one constant. Surface and deep-ocean heat capacities are eight and one hundred watt-years per square metre per kelvin. Ocean exchange transfers heat between the boxes without creating energy. The drawing shows changes in energy flows, with enlarged arrows and temperature colours; it is not a map of regional temperatures. The model omits weather, spatial circulation, changing clouds and ice, carbon-cycle feedback and time-dependent emissions. Its adjustable parameters and trajectories are teaching examples, not fitted observations or climate projections.

Simulated data

## Two layers, two response times

A constant heating step begins at year zero. The moving marker follows the replay.

> Interactive plot of the simulated thermal or magnetic model.

Surface temperature change

Deep-ocean temperature change

## Where the idea came from

[Manabe and Wetherald followed heat through the atmosphere](https://sandboxphysics.com/stories/thermal-complexity/manabe-radiative-convective/) [How the ocean remembers changing weather](https://sandboxphysics.com/stories/thermal-complexity/hasselmann-weather-climate/) [Explore heat, chance and collective behaviour](https://sandboxphysics.com/stories/thermal-complexity/)

Physics tutorial

## A small imbalance can warm a large ocean

**Background** Earth receives sunlight and loses infrared radiation. When these flows balance, its stored energy stops changing. Extra heating disturbs that balance.

**Why it matters** The surface and deep ocean warm at different speeds. Following both explains why a nearly steady forcing can produce a long temperature response.

Start with the essentials

- **Focus question**: What sets the final warming, and what sets how quickly it arrives?
- **One-sentence intuition**: Radiative restoring sets the final warming; heat capacity and ocean exchange set the delay.

### Core mathematical model

#### Heat entering the surface layer

$$
C_s\frac{dT_s}{dt}=F-\lambda T_s-\gamma(T_s-T_d)
$$

The surface receives the extra heating, loses more radiation as it warms, and transfers heat to the colder deep ocean. Time is in years; heat capacities use watt-years per square metre per kelvin.

#### Heat entering the deep ocean

$$
C_d\frac{dT_d}{dt}=\gamma(T_s-T_d)
$$

The same exchange leaves one layer and enters the other. It disappears when their temperature changes become equal.

#### Whole-system energy balance

$$
N=F-\lambda T_s=\frac{d}{dt}(C_s T_s+C_d T_d),\qquad T_{s,\infty}=F/\lambda
$$

A positive restoring coefficient means warming increases energy loss to space. The final surface temperature follows from zero net heat input. It is an equilibrium limit, not a forecast for a particular year.

### Common difficulties

#### Steady heating can still change temperature

**Typical misconception** Once extra heating stops increasing, the temperature should stop rising.

**Better mental model** Temperature stops changing when heat entering and leaving balance. A fixed extra input keeps adding energy until warming has raised the outgoing flow enough.

### Run the experiment

1. 01

   #### Let the ocean take some heat

   Compare The ocean delays warming with Disconnect the deep ocean.

   **What to observe:** Exchange with the deep ocean reduces early surface warming. Both runs have the same eventual surface temperature.

2. 02

   #### Change the restoring response

   Choose More warming from the same heat, then move Cooling response to warming.

   **What to observe:** Weaker radiative restoring allows more warming before the added heat can escape.

3. 03

   #### Reverse the energy flow

   Choose A cooling experiment or drag the gold heating source down.

   **What to observe:** The surface cools first. The deep ocean follows gradually, transferring heat in the opposite direction.

Physical conclusion

The deep ocean delays the response. The radiative restoring response determines the final temperature in this constant-parameter model.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)