Energy Conservation allows Power to Appear seemingly from Nowhere (no, that doesn’t imply unlimited free energy)

Some people are surprised (or even scornful) when they see diagrams of “Earth’s energy budget” (like this one) showing that the Earth absorbs about 240 W/m² of sunlight, and radiates a similar amount to space, but much large power fluxes circulate in the lower atmosphere: on average, the Earth’s surface radiates 398 W/m² upward and the atmosphere radiates 340 W/m² downward to the surface..

Some say “Nonsense! Energy is conserved. You can’t have more power than what arrives from the Sun!” They conclude that those 398 and 340 W/m² numbers must be wrong, and that the scientists who report those numbers must be misguided or lying.

But, scientists do understand basic principles like conservation of energy.

The belief that  energy conservation means “the power can’t be bigger” is wrong, as you can see if you’re willing to take a few minutes to think it through.

Table of Contents

• Power is generally conserved
• Recirculating power can be larger than power entering and exiting
• Water recirculating 
• Charlie Chaplins recirculating
• Energy recirculating in resonators
• Thermal energy recirculating
• Why don’t we sense all that recirculating thermal energy?
• Thermal energy recirculation between Earth’s surface and atmosphere
• Extracted power vs. recirculating power: no “free lunch”
• If you can’t extract recirculating power, why does it matter?
• Recirculating thermal energy and temperature
• Appendix
  • Appendix A: The rules for conservation of power
  • Appendix B: Energy fluxes and discrete power flows
• Postscript

Power is generally conserved

Power is the “rate of flow” of energy. 

Because energy is conserved, you might think that power is conserved also. You’d be mostly right: power is generally conserved. By this, I mean that flows of power can divide and combine, but the total power will generally remain the same.

This “general” conservation of power / flow rate is illustrated in the figure below. (The thickness of each line corresponds to the power / flow-rate of the depicted sub-flow.)

However, there is one circumstance in which power (or any flow rate of a conserved quantity) can behave in a different way than one might expect. 

Recirculating power can be larger than power entering and exiting

When energy is somehow “recirculated”, so that the flow form a “loop”, then power can be present inside the “recirculation loop” that is not constrained by the amount of power entering the system.

This is depicted in the figure below.

This figure will be easier to understand if we look at some concrete examples of “recirculation.”

The easiest way to understand how power or flow rate is affected by recirculation is to consider a few examples involving other things other than energy which are also conserved. 

(Although these conserved things aren’t energy, the math of how conservation works is identical.)

Water recirculating

The quantity of water is conserved (except in circumstances where water is created or destroyed). So, water is a little bit like energy.

Imagine an artificial waterfall in a garden. Suppose 10 gallons per minute flow down the waterfall. There is a pool at the bottom. 0.001 gallons per minute leaks out of the bottom pool, and 9.999 gallons per minute is pumped back up to the top. At the top, a pipe adds 0.001 gallons per minute to replace the water being lost.

So, the flow rate into and out of the system is 0.001 gallons per minute, but the flow rate circulating through the waterfall is 10 gallons per minute, ten thousand times larger.

The rate of water circulating is far larger than the rate of water entering or leaving.

Water is conserved. But that doesn’t put any limit on the flow rate of the recirculating water! 

Charlie Chaplins recirculating

Next, imagine that there are 1000 Charlie Chaplins in the world. They’re not being born, and they’re not dying. So, Charlie Chaplins are conserved. Charlie Chaplins are a little bit like energy.

Suppose there is a house where 1 Charlie Chaplin per minute enters the front door, and 1 Charlie Chaplin per minute exits the back door. 

While inside, each Charlie Chaplin makes three circuits around the inside of the house before exiting. 

In particular, there is a front room and a back room, with two doors between them. During each circuit inside the house, each Charlie Chaplin passes through the right-hand door, from the front room to the back room, then passes through the left-hand door, from the back room to the front room. When he has made his circuit three times, he goes through the right-hand door a fourth time, then exits the back of the house.

So, the flow rate of Charlie Chaplins (CC) per minute in various places is:

1 CC/min into the front door

4 CC/min through the right-hand door

3 CC/min through the left-hand door

1 CC/min out the back door

If you stand at the right-hand door, you’ll see 4 Charlie Chaplins per minute through it.

The rate at which Charlie Chaplins circulate inside the house is larger than the rate at which they enter and leave.

The number of Charlie Chaplins that exist is a conserved quantity, but the rate at which Charlie Chaplins pass by is not. Inside the house, at least, we could increase that rate indefinitely, simply by getting the Charlie Chaplins to take more and more loops between the rooms before they leave.

Energy recirculating in resonators

Energy is a conserved quantity.

Power is the rate at which energy passes by. When energy is recirculating, going from one place to another and back again, the rate of energy passing by (i.e., the power) within the recirculation loop can be larger than the rate of energy entering or leaving.

Does energy ever recirculate in a way that allows this to happen?

* * *

Yes. It happens on an everyday basis: in the “klystron” that powers your microwave oven, in musical instruments, and in lasers. These are all examples of “resonators.”

In any resonator, something bearing energy goes back and forth. In a klystron, it’s microwaves. In an acoustic resonator within an musical instrument, it’s sound waves. In a laser, it’s light.

When any resonator is in use, energy enters the interior of the resonator at a certain rate. But, as that energy bounces back and forth (or recirculates), sometimes the power levels inside the resonator get larger than the level of power that is entering and exiting.

We can experience this directly if we spend time inside an echoey chamber with hard surfaces on all sides—sounds get louder than they would otherwise be.

Thermal energy recirculating

Energy recirculation actually happens all the time, all around us, outside of our awareness:

• Any two bits of matter are continually recirculating thermal energy between them via electromagnetic radiation, unless those bits are separated by something that is capable of blocking that radiation. 

For matter at temperatures tolerable for humans, that electromagnetic radiation will have wavelengths in the range known as “longwave infrared” radiation, or LW IR radiation. 

(People talking about the atmospheric Greenhouse Effect categorize electromagnetic radiation into shortwave (SW) radiation with a wavelength less than 4 microns and longwave radiation (LW) with a wavelength longer than 4 microns. The Sun is hot enough that it emits essentially only SW radiation, and the Earth is cool enough that it emits essentially only LW radiation. Ultraviolet and visible radiation are SW. Some infrared radiation is SW and some is LW. “Greenhouse” gasses are gasses that absorb and emit LW IR radiation.)

All matter (solid, liquid, or gas) is continually emitting radiation with power proportional to σ⋅T⁴ where σ is the Stefan-Boltzmann constant and T⁴ is the fourth power of the matter’s absolute temperature.

(There are nuances to all this, related to materials being better at emitting at some wavelengths than at others, but what I’ve said is accurate enough for our current discussion. For nitrogen and oxygen, the statement that “all matter emits thermal radiation” is true in principle but not so much in practice; nitrogen and oxygen do emit thermal radiation, but with emission rates over 100,000 times smaller than the emission rates of Greenhouse gasses.)

For any two objects, some fraction of the radiation emitted by the first object is absorbed by the second object. (The fraction might be zero, if the radiation is unable to travel between the two objects.) And some of the radiation emitted by the second object is absorbed by the first object. 

What this looks like for two objects at the same temperature is shown below.

The process of emission, travel, and absorption of radiation is symmetric in the two directions. So, the same coupling coefficient, F₁₂, is relevant to the radiation energy flows in both directions.

As you can see from Figure 5, when two objects are at the same temperature, the thermal energy flows back and forth between the two objects. Although there is power flowing in both directions, the power is the same in both directions.

It can be thought of as a closed energy recirculation loop. No power enters or exits the flow. The energy recirculation loop simply exists, perpetually sending energy back and forth.

“Heat” flow is defined as the net energy flow that spontaneously flows between things as a result of their temperature differences.  To obtain the radiation heat flow between two objects, one needs to calculate the net radiation power flow by subtracting the power of radiation traveling in one direction from the power of radiation traveling in the other direction.

So, the rate of radiation heat flow, H₁₂, from the first object to the second object is given by:

H₁₂ = F₁₂⋅σ⋅(T₁⁴ – T₂⁴)

When the two temperatures are equal, the rate of radiation heat flow is zero: H₁₂ = 0

Radiation is perpetually traveling in both directions, and carrying energy with it, and creating a radiation power flow in each direction. Yet, no “heat” is flowing.

What happens when the temperatures are different? This is shown below.

When one object is warmer than the other, the warmer object emits more thermal radiation than the cooler object.

At the level of power flow and energy recirculation, this creates a situation where some power is endlessly recirculating between the two objects, but there is also a portion of the power that flows unidirectionally, from the warmer object to the cooler object.

This unidirectional flow merges with the recirculating flow in the warmer object, and exits the recirculating flow in the cooler object.

The unidirectional power flow is the “heat” flow, H₁₂ = F₁₂⋅σ⋅(T₁⁴ – T₂⁴).

The power of the radiation flows in each direction can be quite a bit larger than the “heat” flow. The size of those flows is set by the Stefan-Boltzman Law, and the requirement that matter always emits thermal radiation with an intensity determined only by temperature.

* * *

Again, any two bits of matter that can exchange thermal radiation is continually and perpetually using radiation to recirculate thermal energy between them. 

And, if they are at different temperatures, there will also be a radiation heat flow from the warmer bit of matter to the cooler bit of matter.

Why don’t we sense all that recirculating thermal energy?

Human senses don’t allow us to directly sense LW infrared radiation.

What we are capable of sensing is heat flow.

If a person with a skin temperature of 35℃/ is naked in a room at a temperature of 27℃/, the coupling factor F₁₂ might be about 1.8 m², and the radiation heat flow from the person’s skin to the surrounding environment would be about H₁₂ = (1.8 m²)×(511 – 460 W/m²) = 94 Watts. 

Since people normally generate heat internally at around that rate, that level of heat loss would likely be pretty comfortable.

That person’s senses wouldn’t be able to tell them that the power of the individual radiation energy flows in each direction are around five times as large as that. All their senses can tell them is how quickly their body is gaining or losing heat.

Yet, while our body’s senses can’t detect the power of radiation flowing in each direction, scientific instruments can.

Or, to be a bit more precise, in this temperature range, scientific measuring instruments typically measure the power of LW IR radiation flowing in a particular direction by detecting heat flows. But:

• By orienting the instrument to receive radiation flowing in one direction or the other, it’s possible to infer the power or radiation flows in each direction.
• The laws of radiative heat transfer apply at all temperatures; at much higher temperatures, scientists do have ways of more directly detecting the power of thermally-emitted radiation flowing in a particular direction.

The model of radiative heat flow is very well-established and verified, and has been in widespread daily use for over a century.

Thermal energy recirculation between Earth’s surface and atmosphere

That law that thermal energy is continually recirculating between bits of matter whenever possible applies, in particular, to Earth’s surface and Earth’s atmosphere.

While the main constituents of air (nitrogen and oxygen) are for all practical purposes transparent to LW IR radiation and don’t emit any, other materials in air (including clouds, “Greenhouse” gasses, and aerosols) are fully capable of participating in the process of thermal energy recirculation.

The more such materials there are in the atmosphere, the more fully this thermal energy recycling process occurs.

It’s not a coincidence that the so-called “Greenhouse” gasses are involved in this thermal energy recycling process. One way of thinking about the “Greenhouse Effect” (GHE) would be to say that the Greenhouse Effect is the phenomenon of thermal energy recirculating as LW radiation between the surface and the atmosphere. (This might be a slightly unconventional definition of the GHE, but it’s ultimately equivalent to other definitions.)

What this looks like is shown in the diagram below, which includes the major energy flows seen in diagrams of Earth’s Energy Budget. (Here is another version of the diagram that appears below.)

Here’s a semi-detailed description of the various energy flows in Figure 7, which I offer for completeness. The following list isn’t essential to the essay and you can feel free to skip it. 

On Earth:

1. Energy from the Sun enters the system (240 W/m²). Some is absorbed in the atmosphere (77 W/m²), and some is absorbed by the surface (163 W/m²). (For simplicity, I’ve omitted Sunlight that ends up getting reflected back to space.)
2. Because of the clouds, Greenhouse gasses, and aerosols in the atmosphere, there is a significant flow of thermal energy recirculating as thermal radiation between the surface and the atmosphere (340 W/m²).
3. Because the surface is warmer than the atmosphere and space, there is also a radiation heat flow from the surface to the atmosphere (18 W/m²).
4. There is a non-radiative heat flow from the surface to the atmosphere (evaporation/latent-heat and thermals/sensible-heat, 105 W/m²).
5. All heat that flows to the atmosphere leaves as a radiation heat flow from the atmosphere to space (200 W/m²).
6. Because space is radiatively like a very cold object (3 K), and because some thermal radiation from the surface is able to reach space (through the “atmospheric window”), there is a radiation heat flow from the surface to space (40 W/m²).

For purposes of this essay, I’ll now shorten all that to saying that, on Earth:

Energy (SW radiation) from the Sun:

1. enters, 
2. gets absorbed, 
3. flows via multiple heat transfer mechanisms, and 
4. leaves as LW radiation emitted to space (240 W/m²).

Thermal energy:

•  recirculates as LW radiation between the surface and the atmosphere (340 W/m²).

Finally, I come back to the issue that had alarmed some people and which motivated me to write this essay:

• The power of the recirculating thermal radiation flow is larger than the power of the heat flow that enters from the Sun and exits to space.

By now, I can offer multiple reasons why that fact shouldn’t alarm anyone:

• Energy conservation permits power to appear seemingly “out of nowhere” as long as that power is inside a recirculating energy loop.
• The phenomenon of recirculating thermal energy is not unique to Earth’s surface and atmosphere; it’s happening continually all around us, between every pair of nearby objects.
• The power of recirculating thermal radiation is set by temperatures, not by the power entering and exiting the system.

Do those reasons make sense to you? I hope so. If not, you might need to review earlier parts of the essay.

That addresses the central issue that led me to write this essay.

Extracted power vs. recirculating power: no “free lunch”

I suspect there’s still a burning question in some people’s minds about all this:

• If power can appear seemingly “from out of nowhere,” doesn’t that mean that it’s a sort of “perpetual motion” machine, and that we could use it to extract unlimited free energy?

Others might object that there is something “fishy” insofar as the reported recirculation rates “count each gallon of water or Charlie Chaplin or unit of energy more than once.”

They are likely thinking, “if I let you count it more than once, you’ll be able to cheat and produce an infinite supply” of energy, or water, or Charlie Chaplins.

So, we’re back to the same issue.

But, the rate at which something can be sustainably EXTRACTED is entirely different than the rate at which something can CIRCULATE.

* * *

In the case of the artificial waterfall, what would happen if we drained water from the bottom at a rate higher than 0.01 gallons per minute? The amount of water circulating inside the system would steadily decrease. Eventually, there would be very little water inside the system. Water would enter at the top at a rate of 0.01 gallons per minute, flow over the waterfall at a rate of 0.01 gallons per minute, exit at 0.01 gallons per minute, and there would be no remaining water that could be pumped back up to the top of the waterfall.

So, even though there was initially a flow rate of 10 gallons per minute inside the system, the maximum rate at which water can be sustainably extracted from the system is only 0.01 gallons per minute.

* * *

In the case of the Charlie Chaplins, what would happen if we tried to force 2 Charlie Chaplins per minute to exit? The number of Charlie Chapllins circulating inside the house would decline, until eventually there would be 1 CC/min entering the front door, 1 CC/min passing through the right-hand door, 1 CCs/min exiting the back door, and no CCs passing through the left-hand door from the back room to the front room.

So, even though there were initially 3 or 4 CC/min circulating inside the house, the maximum rate at which Charlie Chaplins can sustainably be extracted from the house is only 1 CC/min. 

* * *

In the case of Earth, even though there are now 340 W/m² of energy recirculating between the surface and the atmosphere, at the most 240 W/m² of energy could (theoretically) be sustainably extracted from Earth’s energy flows.

* * *

The lesson is that the rate at which something circulates and the rate at which it can be extracted are entirely different things.

When computing flow rates, you count each thing that goes by. It doesn’t matter if you’ve seen it before. 

It’s just that when you’re extracting things, after you count it, it goes away (or perhaps gets used up in some way), guaranteeing that you won’t see it or count it again.

* * *

A circulating power level (or flow rate) is generally larger than the entry/exit power level (flow rate).

But, that does NOT mean that the sustainable rate at which energy (or any other conserved quantity) can be extracted is larger than the entry/exit rates.

There’s no “free lunch” here. High recirculating power levels do NOT lead to unlimited free energy (or free water or free Charlie Chaplins).

If you can’t extract recirculating power, why does it matter?

Some people might question whether those high power levels inside a recirculating energy loop mean anything, given that those power levels are bigger than the rate at which power could be extracted.

Yes, the power levels are entirely real, with regard to their effect on any physical processes that take place inside the zone of recirculating energy.

It’s only if you tried to extract a major fraction of that recirculating power that you would notice anything unusual. In particular, if you were able to extract a lot of power, the recirculating power levels would drop by a larger amount than you might expect.* (In the case of recirculating thermal energy, the only way one could extract that energy would be by using something cold to absorb it.)

Aside from that, power is power.

Footnote:

* In the Charlie Chaplin example, if you went to the right-hand door, where 4 Charlie Chaplins per minute pass by, and if you started extracting 1 CC/min, you’d find that there would no longer be any Charlie Chaplins recirculating, and the flow rate to the door would drop to 1 CC/min. If you were a bit less greedy, and extracted only 1 CC every 2 minutes, then 0.5 CC/min would continue to circle around inside the house three times, leading to a total flow rate to the right-hand door of (0.5 + 3×0.5 CC/min) = 2 CC/min. Even though the original flow rate at that door was 4 CC/min, when you extract 0.5 CC/min, the flow rate drops by 2 CC/min. That’s why I mean by the recirculating flow dropping by a “larger amount than you might expect.”

Recirculating thermal energy and temperature

In the case of recirculating thermal energy in the form of thermal radiation, those power levels directly correspond to the temperature objects inside that recirculation pattern will have when they are in thermal equilibrium.

The higher the radiative fluxes within that region of recirculating thermal energy, the higher the equilibrium temperature will be for objects within that recirculation zone.

Not that, when in equilibrium, these objects are NOT extracting heat (net energy) from that recirculating thermal radiation. They are absorbing and emitting thermal radiation at equal rates.

So, if you do something to increase the amount of thermal energy that is recirculating, you can increase equilibrium temperatures without providing any additional heat flow on an ongoing basis.

Many people find this result surprising, or even unbelievable.

However, I imagine that everyone reading my words has experienced this phenomenon, even though they didn’t think about it in the the way that I’m not describing things.

* * *

Suppose it’s winter. It’s cold outside, and you are in a cabin. You build a fire in the wood stove. The temperature in the cabin rises, but not as high as you’d wish. You notice that there are cold drafts coming through the walls in quite a few places.

So, the next day you go find appropriate materials, you block any chinks in the wall where air might get through.

The next night, you build a nearly identical fire, but that night it’s noticeably warmer inside.

The following day, a friend comes by with a truck-load of tapestries and offers you some. (He’s an odd friend). You’ve solved the problem with cold drafts, but you put tapestries all over the walls, for the art, and for the added insulation.

That night, you build the usual fire, and it’s even warmer inside.

What happened?

Each night, you were supplying heat at the same rate. However, you kept doing things to reduce the efficiency of heat loss from the warm interior to the cold outside.

Reducing the efficiency of heat loss (technically, increasing the absolute thermal resistance, or increasing the insulation R-value of the walls) led to it being warmer inside the cabin,

On successive nights, you were able to increase the temperature inside the cabin, even though you did not change rate of heating from the wood stove.

Heating rate alone does not determine temperature. Temperature is determined by the interplay of heating and cooling. If you make cooling less efficient, that leads to a higher temperature given a fixed heating rate.

* * *

How does that relate to the “thermal energy recirculation” narrative?

“Increasing thermal resistance” and “Increasing the rate at which thermal energy recirculates” are different ways of thinking about the same thing.

If one fills in air gaps that “let in drafts” or adds tapestries to add insulation, this leads to a situation in which the temperature difference between the surfaces surrounding you (the interior surfaces of the cabin walls, or the tapestries) are able to sustain a higher temperature, relative to the air outside, even with the same rate of heat flow.

When those interior surfaces are warmer, in accordance with the Stefan-Boltzmann law, they emit a higher radiant flux of thermal radiation.

As a result, there are higher levels of thermal radiation power circulating between those surfaces and everything in the cabin, including you. You feel (and are) warmer. Even though the power of your heat source (the wood stove) has not changed.

A similar thing happens on Earth when things are added to the atmosphere that interact with thermal radiation. They increase the power of thermal radiation recirculating between the surface and the atmosphere. This corresponds to decreased efficiency of cooling, or increased thermal resistance. The Earth’s surface temperature is higher, even if no additional heating has been supplied by the Sun.

In thermal systems where something supplies heat and that heat flows to a cold thermal reservoir (like space or the air outside a cabin), the temperature isn’t determined by the rate at which heat is supplied alone. Temperature is a function of both the rate of heating and the efficiency of cooling.

The atmospheric “Greenhouse Effect” (GHE) seems like a rather mysterious, even “outrageous,” concept to most people.

I think that may only be because it’s taken a long time to find ways to talk about it clearly.

The way temperatures work in regard to the GHE is the same way that temperature work in every situation in which heat is flowing steadily from one place to another. It’s simply that most people haven’t paid much attention to most of those situations before, or thought carefully about them.

* * *

The situation on Earth, in which the radiative fluxes of LW thermal radiation are individual more powerful than the radiative flux of SW radiation received from the Sun.

That might seem surprising, or suspicious (if you’re inclined that way). But it’s just another example of the countless instances of recirculating thermal radiation happening all around us, all the time.

Appendix

Appendix A: The rules for conservation of power

Feel free to skip this section if by know you already know what you want to know about power conservation.

Earlier, I said that power is “generally” conserved. That’s enough to enter into the subject, but before I conclude, I’d like offer more specific rules for how power conservation works. (These rules also apply to flow rates for other conserved quantities.)

The main rule is:

• Power is always conserved, in the sense that power always flows from somewhere, to somewhere, without changing its amount.

This rule does need some interpretation, however. So, I’ll spell out what it means in some specific situations.

In this essay, I’ve been talking about discrete power flows. By that I mean, if you draw a line through a flow diagram, you can count the number of flows that the line crossed and the count will be a whole number.

For discrete power flows, the power-conservation rule implies:

1. The amount of power in a flow that doesn’t split or merge with another flow will be the same at at every point in the flow.
2. When a flow splits, the initial power will equal the sum of the powers in the flows after the split.
3. When flows merge, the sum of the powers in the merging flows will equal the power in the merged flow.

Every figure in this essay is a diagram of a set of discrete power flows. Each set of flows obeys those three rules.

What catches people off-guard about the power-conservation rule is that:

• Power flowing “from somewhere, to somewhere” can involve power going back to where it began, to form a closed loop.

Surprise about that “recirculation” option is what stimulates the whole “That can’t be right!” response that some people have in regard to the exchange of thermal radiation, or thinking about power levels inside a resonator.

Power flows can also originate, terminate, or accummulate:

• A power flow can originate at a “power source” which just means we are not going to trace where the energy came from prior to that point.
• A power flow can terminate at a “power sink” which just means we are not going to trace where the energy came from prior to that point.
• Power flows can also enter and exit an “energy reservoir”, the power in minus the power out need not be zero, but the difference will determine the rate at which the total amount of energy in the reservoir rises or falls.

In the case of Figure 7, showing Earth’s energy budget, the Sun is the power source, and space is the power sink.

My examples haven’t involved anything functioning as a reservoir. However, reservoirs were present, and would have functioned as reservoirs if the flow rates had been unbalanced. In particular:

• For water flowing through the artificial waterfall, the pool at the bottom is a “reservoir,” and if the flow rate into the system exceeded the flow rate out, the water level would rise. (Figure 3)
• For Charlie Chaplin, the house is a “reservoir,” and if we restricted the flow rate at the back door, Charlie Chaplins would begin to accumulate inside. (Figure 4)
• For heat flow between two objects, the two objects are each “thermal reservoirs,” and an imbalance between the flow rate in and out would result in the total thermal energy contained by the object increasing or decreasing, leading to temperature increasing or decreasing (unless there is a phase change happening). (Figure 6)
• For Earth’s Energy budget, the surface and atmosphere are “thermal reservoirs,” and if we reduced power of thermal radiation reaching space, the thermal energy would accumulate in these thermal reservoirs and their temperature would increase. (This is what happens when the concentrations of Greenhouse gasses in the atmosphere are increased. Again, the Greenhouse Effect reflects a pattern that is not unique to climate, but which occurs all around us.) (Figure 7)

Appendix B: Energy fluxes and discrete power flows

Discrete power flows are an abstraction. In the physical world, power general flows in a continuous “energy fluxes” spread out over some area. Think about the light spreading out from a light bulb.

Thinking about continuous energy fluxes can be tricky for the uninitiated to think about, but I think it’s worth explaining how they can be translated into discrete power flows.

Any field of energy fluxes can be translated into a set of discrete power flows. To do that, one imagines drawing conceptual surfaces through the space where those continuous energy fluxes are flowing. If you want, you can divide the surface into multiple sub-areas that together make up the entire surface.

For each conceptual surface, or subdivision of a surface, total up all the power from the continuous flow that flows through that surface or subdivision of a surface. That total corresponds to the power of a discrete power flow. Then, if you trace how much of the power that flows through one surface or subdivision of a surface flows to a different surface or subdivision of a surface, you have just identified the beginning and end of one discrete power flow.

That may sound a bit abstract and confusing. But, it’s the basis of Earth’s energy budget diagram (Figure 7), for example, as I’ll explain.

• Conceptually, there is a spherical surface just above the Earth’s atmosphere.
  • All the sunlight that crosses that conceptual surface (and is eventually absorbed) is treated as part of the discrete “Sunlight” flow, and the power listed is the total power crossing that conceptual surface.
  • The outbound “LW Radiation” flow is similar, but for LW thermal radiation crossing that conceptual surface from below.
• Conceptually, there is a spherical surface just above the Earth’s surface, below its atmosphere.
  • Power crossing this conceptual surface defines the flows for sunlight absorbed by the surface, non-radiative heat transport upward, LW radiation emitted by the surface, and LW radiation absorbed by the surface.

Any diagram of discrete power flows usually implicitly involves a procedure like this. This procedure allows the complexity of continuous energy fluxes to be summarized in a much more manageable form, as aggregate discrete power flows.

The flows in Earth’s energy balance diagrams like Figure 7 are really discrete flows, even though the values are given in units of W/m², which is a unit also used for continuous energy fluxes. Talking about there being 240 W/m² of absorbed sunlight is more humanly “relatable” than talking about the Earth as a whole absorbing 120 petaWatts (1.2×10¹⁷ W) of sunlight.

Postscript

I hope this essay has been at least a little helpful in demystifying how power levels inside a system can be higher than the power levels that enter and exit the system.

I’ve you’ve found this essay helpful in some way and would like to tell me about it, I’d enjoy hearing what in particular had value for you.

Thanks, and be well.