# How do subgrid-scale parametrisations in climate models work?

Global Circulation Models typically have grids of 100-300km on a side. There are obviously lots of atmospheric processes that happen at smaller scales than this. Convection, cloud formation, the effect of mountains...

How are these processes built in to the model?

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## 1 Answer

In general small scale motions (like convection and formation of local eddies) in geophysical fluid dynamics are treated as turbulence, that is a regime characterized by chaotic motions and rapid, quasi random variations of pressure, temperature and velocity. Those random processes cannot be neglected in boundary layers (layers of flow close to bounding surfaces atmosphere-soil, atmpsphere-ocean) because they averagely involve a flux of momentum and heat from the atmosphere to the soil or to the ocean (or viceversa). In geophysics most notable boundary layers are:

--> The atmospheric boundary layer, that is the bottom part of the atmosphere, about 1000 m thick, in contact with soil and sea surface.

--> The oceanic boundary layer, that is the top layer of the sea, about 10-100 m thick, close to the boundary with the atmosphere.

Hence a climatic large scale model ivolving boundary layers (for example models describing wind driven oceanic circulation) must take into account turbulence average effects: indeed parameterizing small scale phenomena means "taking their average effects into account".

Conversely, far from the boundary layers, turbulence can be neglected. For example global circulation models describing motions of high atmosphere usually neglect turbulence.

The simplest way to take into account the average effects of turbulence is to introduce in the equations of dynamics terms that represent the average friction, for example that described by drag equation:

Fd = - ρ Cd |U| u

where ρ is fluid density, |U| is velocity scale, u is velocity. Fd is called drag force, and is a force by surface unit. It represents the average friction exerted by the atmosphere on the surface. Cd is called "drag coefficient", and can be estimated through experimental observations. Its value can be different in different situations. Drag equation is an empirical relation, and can be deduced by purely dimensional consideretions, like Reynolds number. In particular we can find, using Buckingham theorem, that Cd depends only on Reynolds number.

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