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I've noticed that in many papers it's common to assume the (daily or longer averaged) vertically integrated radiative heating can be expressed $F_z(\text{TOA})-F_z(\text{SURF})$, where $F_z$ is the vertical component of radiative flux, and $\text{TOA}$ and $\text{SURF}$ denote the top of atmosphere and surface respectively, with "top of atmosphere" usually either taken as $z\to \infty$, or as the tropopause height, depending on the context.

I assume this basically reflects the fact that if we express radiative heating $Q$ as a flux divergence $Q=\nabla \cdot (F_x,F_y,F_z)$, vertical integration gives \begin{align} \int_\text{SURF}^\text{TOA} Q \,dz &= \int_\text{SURF}^\text{TOA} \nabla_H \cdot (F_x, F_y) \,dz + \int_\text{SURF}^\text{TOA} \frac{\partial F_z}{\partial z} \,dz \\ &= \int_\text{SURF}^\text{TOA} \nabla_H \cdot (F_x, F_y) \,dz + F_z(\text{TOA})-F_z(\text{SURF}). \end{align}

It seems natural to assume the $\int_\text{SURF}^\text{TOA} \nabla_H \cdot (F_x, F_y) \,dz$ term, which is the net horizontal flux divergence out of the column, will be small compared to the $F_z(\text{TOA})-F_z(\text{SURF})$ term, but does anyone know just how much smaller? What are some reasonable scale estimates for these terms? Are there situations in atmospheric science where net horizontal radiative flux divergence can't be neglected?

For example, I'm imagining a column with a single spherical cloud in it, and the sun directly overhead, but no clouds in any other nearby columns. In such a situation, wouldn't there be a horizontal radiative flux divergence, i.e. a net horizontal radiative flux out of the column? Would this effect still have a negligible impact on net column heating, or does nothing like this occur in real atmospheres?

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  • $\begingroup$ Random thought: yes, though convection is modelled as an adiabatic process, it would mix if there's convection (otoh cloud and sun directly overhead will suppress convection in such stable air with no other clouds around, so very hypothetically), but is the method applicable on small scale ? What about advection, when an airmass exchanges another ? Maybe the scale is the key here ... $\endgroup$
    – user20217
    Jul 14 '20 at 12:52
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Yes, there are horizontal radiation fluxes. These can change the heating rate by 10-40 K hr$^{-1}$. They also change depending on the extent of the atmosphere you may be considering. One can also imagine that they are a bit stronger during the sunrise and sunset, when the sun is not directly overhead and the beam has a larger horizontal path length.

There is work, such as the Neighboring Column Approximation which tries to get around that. There's another paper, but I can't find it right now that also tries to resolve this issue.

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