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344: fix docs for gray atmosphere optics r=szy21 a=szy21

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Co-authored-by: Zhaoyi Shen <11598433+szy21@users.noreply.github.com>
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bors[bot] and szy21 authored Feb 28, 2023
2 parents 72f27f2 + cc266c9 commit e36bd7c
Showing 1 changed file with 7 additions and 6 deletions.
13 changes: 7 additions & 6 deletions docs/src/Optics.md
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The gray atmosphere (more precisely, semi-gray atmosphere) approximates the radiative properties of the atmosphere by taking the optical properties to be independent of wavelength, separately in the longwave and shortwave bands.

### Longwave
Two options are currently supported for computing the optical depth for longwave radiation.
The optical depth (``d``) for longwave radiation, with the "GrayOpticalThicknessSchneider2004" option, follows [schneider2004](@cite):
Two options are currently supported for computing the optical depth for longwave radiation. The optical depth (``d``) for longwave radiation, with the "GrayOpticalThicknessSchneider2004" option, follows [schneider2004](@cite):

```math
\begin{align}
Expand Down Expand Up @@ -43,10 +42,10 @@ d(\phi, p) = \alpha \left[f_l \sigma + (1 - f_l) \sigma^4 \right] \left[ \tau_e
\end{align}
```

where $f_l = 0.2$, $\sigma = \frac{p}{p_0}$ is pressure p normalized by
surface pressure $p_0$, $\phi$ is latitude, and the longwave optical thicknesses at the equator and at the pole are $\tau_e = 7.2$ and $\tau_p = 1.8$, respectively. $\alpha$ is a scaling factor.
where ``f_l = 0.2``, ``\sigma = p / p_0`` is pressure p normalized by
surface pressure ``p_0``, ``\phi`` is latitude, and the longwave optical thicknesses at the equator and at the pole are $\tau_e = 7.2$ and $\tau_p = 1.8$, respectively. $\alpha$ is a scaling factor.

The optical thickness of an atmosphere layer (the differential optical depth) of pressure thickness $\Delta 𝑝$ is
The optical thickness of an atmosphere layer (the differential optical depth) of pressure thickness ``\Delta p`` is
```math
\begin{align}
\tau(\phi, p) = (\alpha * \frac{\Delta p}{p}) \left[f_l \sigma + 4 (1 - f_l) \sigma^4 \right] \left[ \tau_e + (\tau_p - \tau_e) \sin^2\phi \right]
Expand All @@ -57,7 +56,9 @@ The source function for longwave radiation is calculated as ``S = \sigma T^4 / \

### Shortwave

The optical depth (``d``) for shortwave radiation follows [frierson2007](@cite) and [ogorman2008](@cite):
Two options are currently supported for computing the optical depth for shortwave radiation. The optical depth (``d``) for shortwave radiation, with the "GrayOpticalThicknessSchneider2004" option, is 0.

The optical depth (``d``) for shortwave radiation, with the "GrayOpticalThicknessOGorman2008" option, follows [frierson2007](@cite) and [ogorman2008](@cite):

```math
\begin{align}
Expand Down

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