Representations of the atmospheric boundary layer in global climate models

Convective clouds follow one of three general parameterization schemes. The first scheme is moist adiabatic adjustment. The main advantage of this method is that it is simple, in that if the moist adiabatic lapse rate is exceeded, moisture and heat within the vertical layer are adjusted so that air within the layer is saturated.[2][3] In turn the lapse rate is then adjusted to the moist adiabatic lapse rate, conserving energy, raining out excess moisture while not transporting momentum. The disadvantage of this scheme is that it over-estimates convection, as constrains an entire grid box to act in a convective manner, unrealistic in the real atmosphere.[2]

The second scheme is the Kuo parameterization, and is more complex than the moist adiabatic adjustment scheme.[2][3] This scheme is concerned with large-scale moisture convergence as the moisture source for convection. A weakness of this scheme is convective heating, in that this parameterization scheme assumes that convection heats by mixing cloud and environmental air, instead of heating produced by subsidence between clouds.[2]

The Arakawa-Schubert scheme is the third scheme, and is the most comprehensive.[2][3] This scheme models cumulus cloud/environment interactions, entrainment and detrainment of air/moisture, cloud downdrafts, and subsidence outside of clouds. Additionally, this scheme assumes quasi-equilibrium, with cloud dissipation at a "rate sufficient to keep the atmosphere near equilibrium in the face of large-scale destabilization."[2]

There are two methods of parameterizing turbulent fluxes. The first is local closure. Local closure ties the unknown turbulent quantity at a specific point in space to values and gradients of known quantities at the same point. Additionally, local closure likens turbulent transport to molecular diffusion, and is usually first or second order.[1]

The second method of parameterizing turbulent fluxes is nonlocal closure. Turbulence does not just depend on local values and gradients due to the superposition of many individual eddies. Unlike local closure, nonlocal closure links unknown turbulent quantities to known quantities at many points in space.[1]

K-theory (eddy diffusivity/viscosity theory) is a form of local closure, and is the main first order closure scheme within the surface layer. K-theory follows a similar concept as molecular viscosity, in that the turbulent flux of a quantity is proportional to its spatial gradient, with K as the eddy viscosity/diffusivity. K-theory is powerful because the flux of quantities can be directly tied into the gradient of the mean quantities with height, multiplied by a value K. The idea behind K-theory is that turbulence eliminates gradients, creating a net flow of quantities down their gradients.[1] So K is positive.

	Turbulent flux	Gradient
${\displaystyle \tau _{x}=}$	${\displaystyle -\rho {\overline {u'w'}}=}$	${\displaystyle \rho K_{m}\mathrm {d} {\overline {u}}/\mathrm {d} z}$
${\displaystyle \tau _{y}=}$	${\displaystyle -\rho {\overline {v'w'}}=}$	${\displaystyle \rho K_{m}\mathrm {d} {\overline {v}}/\mathrm {d} z}$
${\displaystyle H_{v}=}$	${\displaystyle \rho \,C_{p}{\overline {w'\theta '}}=}$	${\displaystyle -\rho K_{H}\mathrm {d} {\overline {\theta }}/\mathrm {d} z}$
${\displaystyle E=}$	${\displaystyle \rho {\overline {w'q'}}=}$	${\displaystyle -\rho K_{w}\mathrm {d} {\overline {q}}/\mathrm {d} z}$

The eddy diffusivity/viscosity equations are shown in the table above. K_m represents eddy viscosity, while K_H and K_W represent eddy diffusivity of heat and moisture, respectively. τ_x and τ_y represent Reynolds stress (momentum flux) in the x and y directions, H_v represents the turbulent heat flux, and E represents the turbulent moisture flux. ρ is the density of air, u', v', and w' are velocity perturbations, and θ' and q' are potential temperature and moisture perturbations. Other key points regarding K:[1]

• K = 0 when flow is laminar
• K = 0 at the surface
• K increases as TKE (turbulent kinetic energy) increases
• K varies with static stability
• K is non-negative, with turbulent fluxes flowing down-gradient

Diagram of eddy diffusivity

K-theory is best applied in statically neutral environments, with predominantly mechanically-generated turbulence, though it has been applied to statically stable environments.[1] The figure to the right demonstrates how temperature flux works in a stable environment. If a parcel is displaced upward in a stable environment, w' is positive 0, and θ' is greater than the mean θ. So w'θ' is greater than zero. Conversely, when a parcel is displaced downward, w' is negative, and θ' is negative.

Depending on the vertical resolution of the model, K-theory may be used to account for the various fluxes within the ABL.[2][3] Additionally, K-theory can be applied in unstable conditions within the mixed layer if a counter-gradient term is applied, which accounts for buoyancy-driven turbulence.

Bulk aerodynamic formulae

As an alternative to K-theory, ABL fluxes can be accounted for through the use of bulk aerodynamic formulae.[2]

The bulk aerodynamic formulae use C_d, C_h, and C_q, where C_d is the drag coefficient, C_h is the heat exchange (transfer) coefficient, and C_q is the moisture exchange (transfer) coefficient.[3] Each of these coefficients are functions of the known variables, such as the average wind speed at 10m, and average potential temperature and moisture at the surface and 10m. Additionally, each of these coefficients varies based on stability. From the known variables the exchange coefficients can be easily calculated, and thus, the fluxes can be easily calculated as well. The image below shows the equations used for calculating the exchange coefficients and fluxes, with u_* as the frictional velocity.[1]