I have a rather basic mathematical question regarding converting units. I have read some papers that measure a given variable in Joules per meter square (Jm^{-2}) and others that measure the variable in g*cm/cm . Is it possible to convert from one to the other?
http://en.wikipedia.org/wiki/Thermal_stability
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A Joule is $(kg\cdot m^2)/s^2$, so $J\cdot m^{-2}=kg/s^2$. These units don't match up with $g/cm$, so no, it is not possible to convert between them.
On
Generally this is not possible.
$J = \frac{kg m^2}{s^2}$
If we try to convert you units:
$\frac{J}{m^2} = \frac{kg}{s^2} = 1000 \frac{g}{s^2}$
Obviously, squared seconds are not the same as centimetre. But tell us what is the actual variable and maybe we can help. There are cases when some constants are involved. For example, energy is sometimes measured in $cm^{-1}$ and you can convert that to Joules if you multiply the number by Planck constant and the speed of light.
As kaine mentioned in the comment, there might be some charasteristic acceleration involved. If we divide $\frac{g}{s^2}$ by some charasteristic acceleration (for example, gravitational), we can obtain $\frac{g}{cm}$.
The units are not comparable: a joule is $1\dfrac{\text{kg}\cdot\text{m}^2}{\text{s}^2}$, so $1\dfrac{\text{J}}{\text{m}^2}=1\dfrac{\text{kg}}{\text{s}^2}$ has units of $\dfrac{\text{mass}}{\text{time}^2}$, not $\dfrac{\text{mass}}{\text{length}}$, like $1\dfrac{\text{g}}{\text{cm}}$.
Added: After glancing at this paper and this paper, I suspect that if $S$ is the stability index as described in the second paper, which has units of mass over length, then some people prefer to use $gS$, where $g$ is the acceleration due to gravity; this does have units of mass over time squared and appears to have a nicer physical interpretation as work done by the wind per unit of surface area. Numerically the two differ by a constant factor, so either can serve as an index.