Rates of Emission Declines to Reach California’s Goals

Rates of Emission Declines to Reach California’s Goals

Estimates of rates needed to meet climate goals or GHG (greenhouse gases) reductions by 2030 and 2050 have come out.  These seem to be generated by simple mathematical methods, so we will try to make some of them explicit in this article.  Since the calculations are tedious, I will just present the results at the start.

The exponential rates of decline were:

4.34% from 2020 to 2030; then

5.51% from 2030 to 2050; or directly,

5.12% from 2020 to 2050.

The constant rates of decline were:

3.53% from 2020 to 2030; then

3.34% from 2030 to 2050; or directly,

2.62% from 2020 to 2050.

The simplest rates are generated by simple decay models for the emissions.  However, in practice, you don’t just turn things off, but you have to replace them with new energy sources, new energy uses, and new energy distribution networks.  So the models should really be growth models, not declining GHG models.  

Since transportation is the largest GHG source, this would be new renewables, hydro, nuclear, or highly efficient natural gas as resources.  The energy uses would be electric vehicles, plug in hybrids, hybrids, public transportation, or transportation reduction.  Throw into this mix new vehicle batteries, as well as batteries in energy production and storage.

The 30-46% growth in electricity sources involved in electrified transportation would have to be distributed by an expansion in large grid carrying capacity, and workload, all the way down to the local residential or business areas involved.  Those are starting to be modeled at the state level, but only by a minimal $3 million dollar grant.  Each industry involved is probably putting a lot more money into projections, but they need to be conjoined, and only partly proprietary.  At the same time, growing renewables and other clean energy is needed for replacement of other fossil fuel uses.

Anyway, back to the simple declining emission models to start.  We want to first model a linear declining model, as if we have fixed output production lines of energy saving devices.  If we start at time t=0 with N and end at t=T1 years, with to a P percent reduction, or PN, we must reduce N by 

D/T = (N-PN)/T1 per year.

We measure GHG emission in units of MMTCO2e, which stands for Million Metric Tonnes of CO2 equivalent.  A metric tonne is 1,000 kilograms, or about 2,200 pounds, or 1.1 English ton, since a kilogram is about 2.2 pounds.  (It would be much nicer if we called this with a simple common name.  I doubt if any climate scientist would want the unit named for them.  Maybe then it could just be called the ‘havoc’.) 

California decided to define their goals in terms of the GHG emissions in 1990.  (It would have been better if they had started at the scary 2000 Millennium.)  In 1990 California emissions were 431 havocs.  Today, and say at the start of 2020, we are at approximately 400 havocs.  The SB32 goal is 259 havocs in 2030, or a 60% reduction from 1990.  The 2050 goal is 86 havocs, or a reduction of 80% from 1990.  We are just going to calculate from 2020 to these dates, and their havoc goals.

Linear Reduction

Linear reduction is good if you set up the needed set of factories rapidly, but then keep steady employment and output over the period.

For the 2030 goal, D/T = (400-259)/10 = 141/10 = 14.1 havocs a year, linearly.  As a percentage of 400 havocs, that is a 3.525% reduction per year.  

If we went straight to the 2050 goal of 86 havocs from here over 30 years:

D/T = (400-86)/30 = 314/30 = 10.5 havocs a year, or 2.62% reduction per year.

The third option is to use the linear rate to the 2030 goal, and then restart at 2030 with a different linear rate.  At 2030, we are at 259, and reduce to 86, over the next 20 years.  Then

D/T = (259-86)/20 = 173/20 = 8.65 havocs a year, or a 3.34% reduction a year from the 2030 starting point of 259 havocs.

Exponential Reduction

Exponential reduction is useful if we first pick off the low hanging fruit, but then need to invest more time and cost in getting to the more difficult energy conversion tasks.

In this case, the rate of decline of the GHG is proportional to the amount of  GHG still being emitted.  This takes more clean energy conversions at the start, if possible, and fewer as time goes on.  Since we want to get rid of as much pollution as soon as possible, this produces less overall pollution.  Call C the coefficient defining the rate to amount ratio, the equation is:

DN/DT = – C N,

 This is a differential equation for N(T) as a function of time T, and its solution, where N0 is the starting amount, is:

N(T) = N0 exp( -C T), 

where exp is the exponential function, or the number e=2.718 raised to the power ( -C T)

N(T) = N0 e^(-CT).

We see that the starting condition when T=0, e^0 =1, and N(0) = N0.

As T goes to infinity, the exp goes to zero, so the N(infinity) = 0.

If we want to reach N(T1) = N1 at time T1, we can take the natural logarithm of the above equation:

ln( N1/N0) = – C1 T1.  Or, inverting the natural log and changing the sign

ln (N0/N1) = C1 T1.  C1 is then: 

C1 = (1/T1) ln (N0/N1).

2020 to 2030 with Exponential Decline

So, T=0 is 2020, and N0 =  400.  For the leg to 2030, T = 10 years, and 

N1 = 259.  Then C1 = (1/10 years) ln (400/259) = 0.4346/10 years,

C1 = 0.04346/year.  The time dependence is:

N(T) = 400 havocs exp( – 0.04346 T).

The rate of decrease lowers as time goes on, and is the derivative:

DN(T)/DT = – 400 x (0.04346) havocs exp( – 0.04346 T)

DN(T)/DT = – 0.04346 N(T).

So, the decrease ratio is (1/N(T)) DN(T)/DT = -0.04346, or a 4.34% decline each year over the previous year.

2030 to 2050 with Exponential Decline

If in 2030 we start a new rate of exponential decrease to reach the 2050 goal, it could be disruptive.  We evaluate that next, and then look at a smooth decrease from 2020 all the way to 2050.

At 2030, we have N1 = 259, called T=0 again, and in 2050, N2 = 86 with T = 20 years.  Then:

C2 = (1/20 years) ln(N1/N2) = (1/20) ln(259/86) = (1/20) ln(3.0116) = 1.1025/20 = 0.0551, or a 5.51% rate of decline year to year.

Direct 2020 to 2050 with Exponential Decline

We now go from N0 = 400 to N2 = 86, with T = 30 years.

C3 = (1/30 years) ln(400/86) = (1/30) ln(4.6512) = 1.537/30 = 0.0512, or a 5.12% rate of decline, year to year.


The exponential rates of decline were: 

4.34% from 2020 to 2030; then

5.51% from 2030 to 2050; or directly,

5.12% from 2020 to 2050.

The constant rates of decline were:

3.53% from 2020 to 2030; then

3.34% from 2030 to 2050; or directly,

2.62% from 2020 to 2050.

Since the 30 year plan decline rate is less than the 10 year decline from 2020 to 2030, that plan would miss the goals for 2030.

Clearly, a detailed integrated flow plan of hundreds of areas will not be summarized by any simple scheme as above.  The desire for smooth employment will probably push toward linear decline, especially with the higher initial rates of decline required with the exponential decay paths, and then the declines in employment.  However, the linear declines also emitted more greenhouse gases than the exponential curves which lie below them.

In the above cases, the smoother, longer range goals usually gave the lower values. 

In practice, we are in a 8 year period political see-saw, which is devastating to the planet.

If the rich and greedy fossil fuel oilgarchs dominate:  “Cry havoc, and let slip the dogs of war.”  Mark Anthony in Julius Caesar by William Shakespeare.


About Dennis SILVERMAN

I am a retired Professor of Physics and Astronomy at U C Irvine. For a decade I have been active in learning about energy and the environment, and in lecturing and attending classes at the Osher Lifelong Learning Institute (OLLI) at UC Irvine.
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