Can U integrate well even numerically? What I can't but want to.

Discussion in 'Physics & Math' started by Billy T, Feb 25, 2015.

  1. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

    Here is what and why I want to integrate:

    CH4 (methane) is a powerful Green House Gas, GHG, but is more rapidly removed from the air than CO2, mainly by the OH- radical (less than 6% by soil bacteria). That OH- is mainly produced by harsh solar UV high up in the atmosphere. For simplicity, lets consider that only OH- removes CH4 from air.

    For at least 800,000 years the decades long average flux of CH4 into the air was much lower than today and the atmospheric concentration of CH4 remained low (See two section graph at end.) because the production rate of OH- was faster than the release rate of CH4 but that changed a few decades ago with the unprecedented and increasing rate of CO2 release, modern industrial man has made.

    Now the CH4 release rate ( mainly from global warming of frozen tundra's methane ice hydrates, and other man related sources, like oil and gas wells, leaking natural gas distribution pipes and storage tanks, rice culture and many rudiments* raised for food) is higher than the UV limited OH- production rate. Thus, the half-life of CH4 is increasing as the reaction with OH- destroys both reactants.

    For example, in 2003 the CH4 half life was 9.6 years and in 2013 it was 12.6 years. I. e. currently the half-life of CH4 is increasing by one year, every three years and that rate is accelerating as the ratio of CH4 / OH- atmospheric concentrations increases. (Due to both CH4 concentration increasing and OH- concentration falling). I designate the rate of half-life increase as “R(t).” At start of 2015, R(2015) = R(t= 0) = R(0) was 1/3 (years/ per year) so R(t) is a dimensionless number that is increasing with time.

    If R(t) = R = 0 then the half life or 50% Decay time, D, is a constant number of years. Then the number of CH4 molecules remaining after n years, N(n), from when No = N(0) were quickly released at n = 0, is N(n) = No (0.5) ^ (n/ D) and this can be integrated from year zero to year Y, which need not be an integer. Call that integrated result:
    “I(Y).” I(Y) is a bounded function as eventually none of the molecules in the puff remains so I(Y) no longer increases with greater Y.

    CO2 is now the dominate terrestrial cause of global warming changes, even though it is far from the most powerful GHG per unit mass. All other green house gases are compared to it by giving how much (greater, usually) mass of CO2 also released in a “puff” at t = 0 would produce the same global warming during the next few Y years after the puffs were released. This mass ratio, producing equal global warming in the first Y years after puffs are released is called the “Global Warming Potential,” GWP, of the GHG being compared to CO2. As CO2 remains in the atmosphere a very long time compared to many other GHGs, the GWP is almost always a decreasing function of time.

    To calculated the GWP of some GHG, one must also compute the global warming that the same mass of CO2 would produce in the first Y years as the GWP is the ratio of these two masses making same warming in the first Y years after puff release. The removal of CO2 from the air is best evaluated using three different half lives as there are three different removal mechanisms. (Don't get scared off - you can "steal" the CO2 results you will need.)

    Both these “puff decays” (CH4 & CO2) as a function of time have been calculated by repenner at: but with CH4 assumed to have a fixed half-life of 12.4 years, the correct value when he did that. He gave their same effect mass ratios, the GWP for several different years, values of Y.

    With Y very tiny, say only a day, the “Global Warming Potential” of CH4 was 120 times greater than the same mass of CO2 released at t = 0 and for several other periods, Y years after puffs were released;

    The GWP of CH4 during first Y years after puff release (according to repenner) was:
    Y = 0 GWP = 120
    Y= 10 GWP = 104
    Y= 20 GWP = 83.8
    Y =50 GWP = 48.4
    Y =100 GWP = 28.5
    Y =500 GWP = 8.1
    So little CH4 is left after 500 years, that the GWP of CH4 during first 1000 years would be about 4.

    Repenner's results agree well with the relative few years having any values given in the literature.

    My interest is: What would these results be if the half life, D, is not a constant, but is increasing as it in fact does, at least linearly, if not faster as the concentration of OH- falls and that of CH4 increases, but the required integration is too difficult for me to do, even if I assume D(t) just increases linearly with time.

    I. e. lets assume that on 01/01/15 and any later time / date in years, the half life, D(t) = 13 + (t -2015)/3. For example, at start of July 2016, (t – 2015) = 1.5 as t can be with a fraction of a year post 01/01/15. Then, D(1.5) = 13+0.5 = 13.5 years, etc. With this temporally increasing half-life, can you integrate to find I(Y) at least for these five Y values: Y = 10, 20, 40, 50 and 100 years, even by numerical means? I assume you can use repenner's results** for the global warming CO2 makes for all but Y = 40 years. If you don't want to do the Y =40 year case, fine, I'll thank you still. Treating CH4's global warming effect correctly, may be very important. For more on why, see:

    - - - - - - - -
    * Brazil has world's largest beef cattle herd, gets more than 85% of its electric power from hydro-electric dams, (and less than 5% from fossil sources). Brazilians drive mainly alcohol (from sugar cane) powered cars so Brazil’s main release of green house gas is from its vast cattle herd! Anaerobic bacteria live in the cattle and make huge volumes of CH4. Sugar cane alcohol as car fuel is slightly a “negative source” of CO2 as all the carbon coming out of the tailpipe was earlier removed from the air by the growing cane. Some of that carbon is always being stored in car fuel tanks and the distribution system, especially in thousands of large underground tanks at “gas” stations and in ocean transport ships.

    ** I. e. repeat the calculation you did, but for fixed D=12.4 as repenner did and then use his GWP values to discover what the CO2 warming his denominator, with three different Ds, must have been.

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    The historic oscillations correspond to ice ages coming and going. The plotted CH4 concntrations correlate very well with the temperature (both lowest during the ice age) and better than CO2 does. This huge recent spike in CH4, seen in this light is very scary.
    Last edited by a moderator: Feb 25, 2015

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