I would like to elaborate on my previous post, in which I presented a simple temperature vs. time function that causes KMVR2011’s model relating sea level rise rate to global temperature behave in a rather peculiar manner. I am trying to find a balance between simplicity, clarity and thoroughness. The level of mathematical literacy of my readers may vary widely, but this time around I need to employ some calculus. If the equations bother you, just consider the conclusions.
Starting with the conclusions
There exists a simple class of realistic temperature vs. time functions, which when applied to KMVR2011’s model yield results that disqualify it as representing a relationship between global temperature and sea level rise rate. This class of temperature vs. time functions gives a family of curves for which it is guaranteed that the higher the temperature the lower the sea level rise rate. This implausible effect is so severe that if forces rejection of the KMVR2011 model.
Here is the KMVR2011 model
Where H is the sea level, T(t) is the global temperature, Too, a1, a2, b and τ are all constants and To(t) is a to-be-determined time varying function related to T(t) as defined by equation Ia.
Now, consider the following temperature evolution. It is nearly the same as equation II from my previous post, but has an additional unitless constant, γ (a.k.a. “gamma”), in the exponential…
If equation II is inserted into equation I, then…
Rearranging terms in equation III gives…
H is the sea level. dH/dt, the derivative of the sea level, is the sea level rise rate. d2H/dt2, the second derivative of the sea level, is the rate at which the sea level rise rate changes. That is, d2H/dt2, is the acceleration. If d2H/dt2, is positive, the sea level rise rate is increasing. Conversely, if d2H/dt2, is negative, then the sea level rise rate is decreasing. Taking the time derivative of equation IIIa gives…
Let’s also consider the difference in the sea level rise rates at some time, t, for different values of γ. We can do this by analyzing the derivative of dH/dt (equation IIIa) with respect to γ.
What does the math tell us?
KMVR2011 does not conclude with specific values for their model constants and their time varying T0(t). Instead, they present probability density distributions for some constants, or combination of constants. However, there are some definite constraints that can be noted about the variables and their relationships to each other. These constraints are key to my conclusions.
a1 + a2 = a, where a1 and a2 are defined in KMVR2011 (see equation I, above) and a is defined in VR2009. VR2009 found a = 5.6 mm/yr/K.
a1 > 0 mm/yr/K and a2 > 0 mm/yr/K. KMVR20011 states that the distribution of a1 for their Bayesian analysis varied between 0.01 and 0.51 mm/yr/K. Needless to say, if either of these terms were less than zero the KMVR2011 model would make even less sense that it does now. That would be a road that the KMVR2011 authors do not want to travel.
C > 0. C is a unitless constant that I introduced, and for the purposes of this post I am constraining C to be greater than zero.
γ > 0. γ is a unitless constant that I introduced, and for the purposes of this post I am constraining γ to be greater than zero.
Time, t, is restricted to about 1900 and later for my hypothetical temperature (equation II). This insures that T(t) > T0(t), which in turn insures that dT0(t)/dt > 0.
The equations above, coupled with the listed constraints guarantee the signs of the derivatives shown in table 1, below.
Table 1. Derivatives of temperatures and second derivatives of sea level. Green “up arrows” indicate increasing values and red “down arrows” indicate decreasing values.
As you can see from table 1, it gets little confusing for 0<γ<1. When a1, a2, b, C, and γ conform to the listed constraints, the signs of the various derivatives are known with certainty as long as…
But when …
at some point in time t- t’ will become large enough that d2H/dγdt will become positive. When that time occurs depends on the choices of a1, a2, b and γ. If we choose a1, a2 and b to agree with VR2009 (recall a1 +a2 = a = 5.6 mm/yr/K, and b = -49 mm/K) and γ = 0.8, then d2H/dγdt will continue to be negative until t – t’ = 44 years.
The conclusion, again.
Equation 2, above, can be used to build realistic hypothetical temperature evolutions. See figure 1, here, for some examples. Remember, KMR2011’s model relates sea level rise to temperature, and when applied to these hypothetical temperatures it must yield realistic sea level rises. It does not.
Table 1 shows various aspects of temperature and sea level using my hypothetical temperature evolution and KMVR2011’s resulting sea levels. Summed up succinctly, the table shows that with this combination the greater temperatures result in lower sea levels. This implausible situation disqualifies KMVR2011’s model.
I realize that a bunch of equations and a table do not give visceral understanding of this effect. A graphical illustration of these points will be coming soon.
Table 1 corrected. Change makes no difference to conclusions.