Smooth transition at thresholds
To avoid abrupt change of values at thresholds we can use the logistic function to make the transition smooth. The logistic function is defined as
\[f(t, k) = \frac{1}{1+e^{-kt}}\]
where
- $t$ is the independent parameter,
- and $k$ is the growth rate of the curve characterizing the steepness of the transition.
The value of the logistic function changes from zero at $t \rightarrow -\infty$ to one at $t \rightarrow \infty$ with the majority of the change happening around $t = 0$. In our microphysics applications, the independent parameter (typically specific humidities) takes only non-negative values, and when this parameter is zero the function should return zero. Thus, we use the change of variable $t = x/x_0 - x_0/x$, where $x_0$ is the threshold value of $x$. Therefore, the logistic function for smooth transitioning from $f(x) = 0$, for $x < x_0$, to $f(x)=1$, for $x > x_0$, is defined as
\[f(x, x_0, k) = \frac{1}{1+e^{-k(x/x_0 - x_0/x)}}\]
Note that when both $x$ and $x_0$ are zero the value given by the above equation is undefined. In this case we return a zero value instead.
Smooth transition of derivative at thresholds
When the function itself is continuous but its derivative changes abruptly at a threshold we can use the indefinite integral of the logistic function for smooth transitioning. In this case, the following function can be used
\[f(x, x_0, k) = \frac{x_0}{k} \ln\left(1+e^{k(x/x_0 - 1 + a_{trnslt})} \right) - a_{trnslt} x_0\]
where $a$ is a fixed value that is defined to enforce zero at $x=0$:
\[a_{trnslt} = -\frac{1}{k}\ln\left(1 - e^{-k}\right)\]
The curve defined by the above equation smoothly transition from $f(x) = 0$, for $x < x_0$, to $f(x)=x-x_0$, for $x > x_0$. Note that when both $x$ and $x_0$ are zero the value of the function is undefined. In this case we return a zero value instead.
Example figures
include("plots/Thersholds_transitions.jl")"/home/runner/work/CloudMicrophysics.jl/CloudMicrophysics.jl/docs/build/N_d_B1994.svg"
