Tipping points & feedback loops

To understand more about tipping points, consider the tipping point of a tall block, standing on its square base, on a table. After the block tilts past its tipping point, the below “self-amplifying feedback loop” becomes dominant and escalates the toppling of the block.

The concepts of “tipping points” and “self-amplifying feedback loops” are relevant to climate change. If we humans continue heating our planet, we will push our climate past a tipping point, triggering a self-amplifying feedback loop that will escalate global heating. This heating could push our climate past a cascade of other tipping points and destroy the climate that has nurtured us.

I write using no mathematics, except for a hidden section that details the mathematics of a toppling block or brick. You can view this maths if you want. The maths informs this discussion about “climate tipping points” and the linked “climate self-amplifying feedback loops” that threaten our climate. These loops are often referred to as “positive feedback”.

I would have written about a “wine glass tipping on a table” but the mathematics / physics of a “toppling brick” is simpler.

For those who venture into the maths section, you can see the differential equation describing the motion of the toppling block.

The movement of the block

When you push the top of the block with your finger, so that it tilts, it will rotate on the edge that stays on the table. It will lean more and more, like the leaning tower of Pisa, until it reaches its tipping point. After you push the block just beyond its tipping point, the weight of the block makes it unstable. The block will topple, rotating around the edge on the table, until it hits the table. There are several stages in this:

State 1: The block standing on its base

Initially, the block is standing tall on a table. When you push it a little with your finger, and then remove your finger, the block will return to standing. Gravity causes this stability as it holds the block against your finger as you withdraw your finger, so the block moves with your finger back to standing.

Small disturbances will not destabilise this block, so this is a state of stable equilibrium.

State 2: The finger-tilted block

As your finger pushes the block onto an angle, you are supporting some of the block’s weight with your pushing finger. It takes less and less pushing to move the block as more and more the block supports its own weight on the edge of the block in contact with the table.

State 3: The tipping point

At the tipping point, the block comes into balance and you no longer have to push the block at all to keep it tilted. In theory, you could remove your finger and the block would stay there, balanced all by itself.

However, the block is in a state of delicate balance, and the slightest vibration or wind will destabilize it. It is in equilibrium, but this is an unstable equilibrium.

State 4: The toppling block

Once your finger pushes the block past the tipping point, the block becomes over-balanced and it starts toppling, rotating on one edge.

State 5: The toppled block

The block stops rotating when it hits the table. The fall can be destructive as the block can break if it is fragile, like a wine glass. If the block is heavy enough, the table can break.

Relating the toppling block to climate change

As we can understand the motion of a toppling block in mathematical detail, this may help us understand “climate tipping points” and the linked “self-amplifying feedback loops”.

The following conclusions about climate change assume that these processes have similarities to the dynamics of the toppling block.

There are many climate-threatening feedback loops. Here I mostly use as an example the amplifying feedback loop escalating both “global heating” and the “release of methane from melting tundra”.

The initial disturbance and control

For the block, the initial disturbance is the “pushing finger” which increases the tilt of the block. Initially, when you withdraw the pushing finger, the block will follow the finger and the tilt reduces. So, at first, the initial disturbance is the controlling factor.

For our climate, the initial disturbance or “pushing finger” is the human burning of fossil fuels and forests. This has greatly elevated the levels of CO2 in the atmosphere and caused a build-up of heat on our planet, global heating. The planet is heating at a rate of:

Four Hiroshima bomb detonations worth of heat every second
22 electric kettles running at 1500 Watt, 24 hours a day, for each of the 7,700 million people on the planet
2.5 multiplied by 10 raised to the power of 14 joule/second

Scientists are urging us to reduce global heating by reducing carbon dioxide levels. They still regard this carbon dioxide level as the controlling factor but are concerned that this may change.

Diagram of the toppling block

The diagram shows a side view of the toppling block. It shows the height and width of the block in a cross-section that passes through the center of the block, halfway into the depth of the block.

The mathematics section

***** Click to view, or hide, the mathematics section.

Viewing the diagram

You might like to view the diagram as you read this maths section. To do this, you may be able to open this page twice, and place:
. the above diagram of the block on the right of your screen, and
. the maths on the left.

Variables, Notation & Terms

Here is some mathematics describing how the block rotates, from after it passes its tipping point until it hits the table.

I present the general case using variables, e.g. using H for height. I also present one example, e.g. height = 4 metres.

H: Block Height = H metre = 4 metre
W: Block Width = Depth = W metre = 1 metre
M: Block mass = M kilogram = 10 kilogram
g: Acceleration due to gravity = g = 9.8 metre per second, per second
T or Tilt: The pushing finger on the top-left edge of the block at E pushes to the right, so the block tilts, rotating clockwise around the edge on the table at A. The angle of tilt away from vertical of the block = Tilt = T degrees. Initially, the tilt is zero and after the block has fallen, the tilt is 90 degrees.
The “landing side” is the side of the block on which it finally lands, length H (Line AD).
The “top of the block” is the uppermost side of the block when it is standing tall, length W (Line ED).
The “diagonal” is the diagonal of the block that passes through the axis of rotation at A (Line AE).
The “axis of rotation” is the edge of the block which stays on the table and about which the block rotates (Edge at A).
C: The “centre of mass” of the block is at point C.
The “wrench” is the line running from the centre of mass to the axis of rotation (Line AC). A wrench is a plumber’s tool. With a wrench, a plumber exerts force via the “wrench handle” to turn a “nut”. For the block, the “wrench handle” is the centre of mass, as the force of the weight acts through the centre of mass, and the “wrenched nut” is the rotating block itself.
F: The force of the block’s weight acting on the centre of mass
I: The moment of inertia of the block
Q: The torque or twisting force on the block, exerted by the weight force
s: The “strength angle” is the “tilt past the tipping point”: As this angle increases, the weight force increases the strength of its twist on the block: s degrees.
u: the “unstable equilibrium angle” or “tipping point angle” is u degrees.
The Y-axis is vertical and the X-axis is horizontal.

Mathematical notation:

The * in equations indicates multiplication
Square root of 2 = Sqrt(2)

The centre of mass

The centre of mass (COM) of the block is in the middle of the block. It is at the point (C), halfway up the height, halfway along the width and halfway down the depth. This is where the two diagonals met, halfway along the diagonal EA. (This assumes that the weight of the block is uniformly distributed throughout the block.)

As the block rotates, the centre of mass travels in a part-circle around the axis A, with the radius being the length of the wrench, R metres

Pythagoras gives the length of the diagonal of the block:
Diagonal = Sqrt( W * W + H * H ) = 4.12 metre

The length of the wrench is half the diagonal
Radius = R = 0.5 * Sqrt( W * W + H * H ) = 2.06 metre

Diagram of the block standing on the table

the block standing tall showing the tipping point angle

The tilt at the tipping point

From the above diagram, notice that as the block rotates clockwise through u degrees, the initially vertical “landing side” rotates and the “diagonal through A” rotates until this diagonal becomes vertical. At this point, both the diagonal and the “centre of mass” are directly over the supporting edge of the block at A. This is the tipping point.
Tipping point tilt = u degrees

When you set the width and the height, you also set the angle u. Consider the right angle triangle AED:
Tan(u) = opposite side / adjacent side = W / H = ¼
So u = 14.0 degrees

The clockwise twisting force: Torque (Q)

The block’s weight exerts a force on the block (F)
F = M * g = 10 * 9.8 = 98 Newton

The angle between the wrench and the vertical weight force is s degrees. Angle s is also the “tilt beyond the tipping point” as
s = T – u.

The block rotates about the edge A, with a part of the weight force causing the rotation. That is the part that acts at right angles to the wrench = F * Sin(s).

The clockwise twisting force or torque (Q) is equal to the length of the wrench (R) by the component of the weight force acting at right angles to the wrench.
Torque = Q = R * F * Sin(s)
So, Q = R * M * g * Sin(T – u)

Clockwise Torque Q = 202.03 * Sin( Tilt – 14)

How Torque changes with tilt.

Graph of torque versus tilt for the toppling block

The graph shows that:

The torque is zero at Tilt = 14 degrees. This is the tipping point and it makes sense, as there is no twisting force when the block is supporting its own weight at the tipping point.
The Torque keeps on increasing from zero as the Tilt increases from the tipping point at 14 degrees to when the block has fallen at Tilt = 90 degrees.
The clockwise torque is negative, i.e. anticlockwise, from when the block is standing on its base at Tilt = 0 until the block reaches the tipping point at Tilt = 14 degrees. This makes sense as, for these tilts, the weight force exerts an anticlockwise twisting force on the block and the pushing finger works against this anticlockwise torque as it tilts the block.

Moment of Inertia

As an object’s mass increases, the force required to move it increases. In a similar way, as an object’s “moment of inertia” increases, the twisting force required to rotate it increases.

from above, we have the torque applied to the block (Q). By working out the “moment of inertia” of the block (I), we can work out the angular acceleration (A) because:
Q = I * A

The “moment of inertia of the block for rotation around the edge A” equals “the moment of inertia for rotation around the centre of mass” plus the “mass of the block” multiplied by the square of “the distance of the centre of mass from the edge A”.

Moment of inertia for rotating the block around the centre of mass
= M * (W * W + H * H) / 12
(See Wikipedia)

The moment of inertia of the block for rotation around the edge A is I:
I = M * (W * W + H * H) / 12 + M * R * R
So, I = 1.25 * M * R * R = 56.67

Acceleration of the tilting

Acceleration of the tilt: A = Torque / Moment of Inertia
So, A = Q / I

Q = R * M * g * Sin(T – u)
I = 1.25 * M * R * R = 56.67

A = 0.75 * g * Sin( T – u) / R

A = 3.57 * Sin ( T – 14) degrees per second per second

Acceleration of the tilt equals the torque divided by 56.67, so it follows the same pattern as the torque in the above graph. As the tilt increases from 14 degrees at the tipping point, the angular acceleration also increases.

Differential equation

A = 3.56 * Sin(Tilt – 14)
Now tilt-acceleration = A = d/dt (d Tilt/dt), so
d/dt (d Tilt/dt) = 3.56 * Sin(Tilt – 14)

This is a second-order non-linear differential equation involving Tilt and time.

Differential equations similar to this could be used to model other feedback loops. Before I worked through this, I often wondered about how maths would represent feedback. Of course, being able to model the toppling block barely scratches the surface of knowing how to model the complex climate feedback loops like the Tundra methane self-amplifying feedback loop.

A reader has suggested a book about the general structure of tipping points: “Catastrophe Theory” by V.I.Arnold.

Solving the differential equation

I wrote a computer program to solve this equation for the initial conditions:
. Time = 0 seconds
. Tilt = 35 degrees, i.e. 21 degrees past the tipping angle, and
. Tilt Speed = 0 degrees per second, i.e. initially at rest
. Tilt acceleration: as the differential equation makes clear, the tilt-acceleration only depends on the “tilt”, so the “initial tilt” of 35 degrees sets the “initial tilt acceleration”.

Then you can calculate what happens each time the block tilts say 0.5 degrees further. For each step change of the tilt you know the start time, start tilt, start tilt-speed, start tilt-acceleration, the next tilt and, via the differential equation, the next tilt-acceleration. From this you can calculate the average tilt-acceleration over the step. As an approximation, you can take this “average tilt acceleration” as the “constant acceleration” over the step, and so calculate the next tilt-speed and the time taken for the step.

By repeating this stepping process until the block hits the table at tilt = 90 degrees you end up knowing in detail how the block has toppled. For each time considered you have, the time, tilt, tilt-speed and tilt-acceleration. You have “solved” the differential equation for the given initial conditions.

Note, the discussion here is in degrees, however, when doing the calculations use radians for angles, radians/seconds for speed, and radians/seconds quared for acceleration.

The toppling time for this block, given these initial conditions, is 1.07 seconds. If you need greater accuracy you can use smaller steps in tilt.

You can also calculate “rotational energy” of the block for each time considered, as plotted below.

Rotational energy of the toppling block

If you want to stop the toppling block, say by catching it, you need to counter the rotational energy that the block acquires as it topples.

Let w = the angular speed of the block in radians per second
Let I = the moment of inertia of the block – see above

The rotational energy of the block = 0.5 * I * w * w

Graph of the kinetic energy of the toppling block versus time

This graph of the kinetic energy of the rotating block shows how the kinetic-energy increases slowly for the first 0.4 seconds, and then increases explosively. Now, the greater the rotational energy, the more energy must be put in to stop the block toppling. So if you need to stop the toppling, it is important to stop it as soon as possible.

The impact of changing the size of the block

The angular acceleration = 0.75 * g * sin(tilt past the tipping point) / R

So the angular acceleration, and thereby the toppling time of the block, depends on “R”, the radius of the movement of the centre of mass. As you increase R, you decrease the toppling time. Now, R = 0.5 * Sqrt(H*H + W*W), so increasing the height or the width of the block will decrease the toppling time equally.

Also note that the mass of the block is not included in the equaiton and so does not influence the toppling time.

If you increase the height, while keeping the width the same, you make it easier to topple the block because the “tipping point tilt” decreases, and you have to tilt the block less to reach the tipping point.

End of the maths section

The acceleration of the tilt increases as the block topples

There are three measures concerning the tilt to consider here:

The tilt, measured in degrees,
The tilting speed, i.e. the rate of change of tilt over time, measured in degrees per second, and
The acceleration of tilting, i.e. the rate of change of the tilting speed over time, measured in degrees per second, per second.

As the block tilts further past its tipping point, more of the weight acts to topple the block so the twisting force increases. Physics dictates that as the force increases, the acceleration also increases. So, as the tilt of the toppling block increases, the acceleration of the tilt also increases.

Textbooks seem to keep their examples simple by mostly considering constant acceleration. I found it interesting that for the toppling block we have increasing acceleration.

Amplifying feedback takes over after the tipping point

The more the block tilts, the more the torque increases, and the block accelerates into the fall, with increasing acceleration. The more it tilts the more it is going to tilt. After the block passes its tipping point, an amplifying feedback loop becomes dominant, we have:

an increase in tilt tends to
increase the twisting force which tends to
increase the acceleration of the rotation which tends to
increase the tilt.

Diagram: The self-amplifying feedback loop that is dominant as the block topples

For a long time, I have wondered about how tipping points and amplifying feedback loops are related, and here we have a clear relationship between a tipping point and an amplifying feedback loop, i.e. after the tipping point, an amplifying feedback loop becomes dominant.

After the tipping point, a new controlling factor emerges

When you move past a tipping point, control over the system changes as an amplifying feedback loop becomes dominant.

For the block, the initial disturbing factor is the pushing finger, something that should be easy to control. After the tipping point, gravity takes control via the amplifying feedback loop of “more tilt causing more twisting force, causing more tilt”.

For our climate, the initial disturbing factor is the carbon dioxide concentration in the atmosphere, something that humans could limit. After the tipping point, a factor like methane release could take control, via an amplifying feedback loop of “more heating causing more methane release from frozen Tundra, causing more heating”.

Regaining control after passing a tipping point

To stop the block crashing after it passes its tipping point, you have to stop continued pushing by the finger. You also have to do something to counter gravity, the new controlling factor, something like catching the block.

To limit global heating after passing a climate tipping point, you have to eliminate the initial disturbing factor, the high levels of CO2 in the atmosphere. On top of this, you have to find a way of countering a new heating force. If this new heating force were the release of vast amounts of methane from melting tundra, then we would be facing a very difficult and dangerous problem.

Here I focus what would happen if the block is gently taken just past its tipping point. What is happening to our climate is very different. There is little sign that humans are inhibiting the “controlling pushing finger” of the greenhouse gases in our atmosphere. We are not gently taking our climate to just past its tipping point. We are sweeping the climate wine glass off the table by prolonged drunken debauchery.

Cascade of feedback loops

When the toppling block hits the table, there could be a cascade of repercussions, e.g. if the block was heavy, the table could be stressed past its breaking point and then even the floor could be stressed past its breaking point.

When considering climate, the escalation of gobal heating due to one feedback loop, like methane release, could lead to the climate pushing past the tipping point of other amplifying feedback loops leading to a cascade of these heating dynamics.

This is the big danger of the current global heating. It could push the climate past one tipping point and this could then lead to a cascade of other climate tipping points.

Work needed to get to the tipping point

The block is initially in a stable equilibrium and the pushing finger has to do work to elevate the block to the tipping point, before the block topples.

For the methane feedback loop, the work is the heating needed to melt the Tundra ice and release the methane. The current high levels of greenhouse gases are currently heating the planet and doing this work. We are already seeing melting Tundra and methane release.

The duration of the motion

For the block, the action occurs in seconds. Our complex climate system is quite the opposite, with the action played out over thousands of years. Significant heating is soon evident in increased atmospheric temperatures and increased ocean surface temperatures. However, some impacts of heating take time. For example, the melting of ice occurs slowly as it takes large amounts of heat to melt ice. This means that increased sea levels due to melting ice will become evident slowly. Also, the deep ocean currents move slowly, redistributing heat around our vast oceans.

The reversibility of feedback loops

The feedback loop of “more tilt leading to more tilt” is not reversible. In a shortened form this loop is (1) an increase in tilt tends to (2) increase the twisting force, which tends to (3) increase the tilt. Reversing the first element of this loop you have a decreased tilt, which does decrease the twisting gravity force. However, there is still a gravity clockwise twist, so the tilt will still increase, not decrease. So this amplifying loop governing the toppling block will not run in reverse.

Many of the self-amplifying feedback loops linked to climate change are also irreversible. For example, the methane loop cannot be reversed. Once warming releases frozen Tundra methane into the atmosphere, subsequent cooling will not capture the methane, freeze it, and return it underground.

However, some climate feedbacks are reversible. For example, the “warming leading to decreased Arctic sea-ice” loop is reversible as cooling can create more ice which can lead to further cooling.

Identifying the tipping point

For the block, the tipping point occurs when the centre of gravity of the block moves directly above the supporting edge of the block.

While science makes it clear that our planet is heating, climate tipping points are not clear cut.

For Tundra methane release, the tipping point would occur when “the annual heating impact of increasing atmospheric methane” equals “the maximum annual cooling impact of reducing atmospheric CO2 levels”. This is not a very useful concept as we are still increasing atmospheric CO2 levels, seemingly unable to reduce CO2 emissions, let alone reduce the CO2 level in the atmosphere.

Conclusions

This analysis suggests a relationship between tipping points and self-amplifying feedback loops that may be general, i.e. after passing a tipping point, a self-amplifying feedback loop gains control of the process.

Once we hit the climate tipping point associated with, for example, the release of methane, then a feedback loop amplifying the methane release could become dominant – and we could expect a cascade of other climate tipping points to be breached. We must act with the greatest urgency to reduce greenhouse gasses in the atmosphere and hope that we have not already passed one of these tipping points.