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I am trying to solve the following equation numerically $$|u_\text{max}|=\frac{\Delta p}{|f|\rho}\frac{\sqrt{2}}{R}\mathrm e^{-1/2} \tag{1}.$$ Here, $\Delta p=20\ \mathrm{hPa}$, $R=500\ \mathrm{km}$ and $\rho=1\ \mathrm{km/m^3}$. For equation $(1)$, $f$ denotes the Coriolis parameter, which in this case equals $$f=2\Omega\sin(45).$$ Using Wikipedia, the SI units for $\Omega$ are in $\mathrm{rads/s}$. Does this mean that the SI units for $f$ are also in $\mathrm{rads/s}$? Being the velocity, the SI units of $u_\text{max}$ should be in $\mathrm{m/s}$. Substituting all values into $(1)$,

$$|u_\text{max}|=\sqrt{\frac{2}{\mathrm e}}\frac{20\times 10^2 \ \mathrm{Pa}}{\Omega\sin(45) \ \mathrm{rads/s}\times 1 \ \mathrm{kg/m^3}\times \left(500\times 10^3\right)\ \mathrm{m}}.$$

I'm unsure of how the correct SI units ($\mathrm{m/s}$) appear.

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The mentioned "$\mathrm{rads/s}$" is not a correct SI unit symbol. (Note that the Wikipedia page on Coriolis frequency also shows wrong units symbols for the hour "$\mathrm{hr}$" and minute "$\mathrm{m}$".) The the correct special symbol for the radian is $\mathrm{rad}$. The radian is a special name for an SI derived unit. It can be expressed in SI base units as follows.

$$1\ \mathrm{rad}=1\ \frac{\mathrm m}{\mathrm m}=1$$

Therefore,

$$1\ \mathrm{rad/s}=1\ \mathrm{s^{-1}}$$

Furthermore, you should know that $1\ \mathrm{Pa}=1\ \mathrm{kg\ m^{-1}\ s^{-2}}$; thus

$$\frac{1\ \mathrm{Pa}}{1\ \mathrm{s^{-1}}\times1\ \mathrm{kg\ m^{-3}}\times1\ \mathrm m} =\frac{1\ \mathrm{kg\ m^{-1}\ s^{-2}}}{1\ \mathrm{s^{-1}}\times1\ \mathrm{kg\ m^{-3}}\times1\ \mathrm m} =1\ \mathrm{m\ s^{-1}}$$

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Anything with radians or lengths at right angles should be treated with care and caution.

SI is a mess for angles, which reflects issues in the wider community about levels of abstraction.

For instance the unit Hertz is just "per second". It can be anything per second. Because it can be confused there are even explicit notes in the bipm web site (owners of SI) about the problem and how they needed extra 'units' for radiation, which otherwise would also have units of Hertz.

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