For demonstration purposes, below is the LTspice simulation of your noise file feeding a precision full-wave op amp rectifier, whose output is filtered by a simple 1-pole low-pass RC filter with a 2 second time-constant.

The green trace is the output from your wav file, the red trace is the rectified output, and the yellow trace is the filtered output.
As you can see, the filtered output reaches near the steady-state value after about 3 time-constants (6 seconds) from a step change in input.

Is that sort of the response you are looking for?

Note that the circuit is basically for demonstration only, and would likely be implemented somewhat differently in a real circuit.

 

I have a 0-10V DC signal representing a scalar value measuring some machine noise. This fluctuates wildly (+\- 2V) however if the values were recorded and a moving average was applied (plot point X is the sum average of the preceding N samples) then what is produced is a smoothly increasing or decreasing trend line.

If I understand the above statement...the input is 0-10 VDC, and it fluctuates by +/-2vdc at some nominal voltage between 0-10v.

I've shown a sample-and-hold circuit, below, that uses 2-LF398 Sample-and-Hold chips and a CD4093B chip.
I've also shown a 10 VDC input voltage that rapidly fluctuates between 8VDC and 12VDC. Initially, U1 samples the varying input signal, while U2 holds the final sampled value. U3 forms a variable frequency clock generator that can be adjusted for a sample time of (Ton) 0.37s to 1.1s.

 

Since, in post #9, you stated that the noise was there for 200ms and then off for 2.8s, I added an analog switch (U3) to average the rectified noise for 200ms and then hold the filtered value for 2.8s.
Below is the LTspice simulation of that.
I shortened the off time to 300ms to speed up the simulation.
In the real circuit you would need to synchronize the switch times with the presence and absence of the noise.

Is that more or less what you are looking for?