Surface fluxes

The surfaces fluxes in SpeedyWeather represent the exchange of momentum, heat, and humidity/moisture between ocean and land as surface into or out of the lowermost atmospheric layer. Surface fluxes of momentum represent a drag that the boundary layer wind experiences due to friction over more or less rough ground on land or over sea. Surface fluxes of heat represent a sensible heat flux from a warmer or colder ocean or land into or out of the surface layer of the atmosphere. Surface fluxes of humidity represent evaporation of sea water or condensation of humidity onto the ocean surface. Similarly, humidity fluxes over land with a given soil moisture and vegetation's evapotranspiration.

Surface flux implementations

Currently implemented surface fluxes of momentum are

using SpeedyWeather
subtypes(SpeedyWeather.AbstractSurfaceMomentumFlux)

1-element Vector{Any}:
 SurfaceMomentumFlux

Interdependence of surface flux computations

model.surface_condition extrapolates atmospheric variables (wind, temperature, humidity) to the surface. model.boundary_layer computes centrally a drag coefficient for surface momentum, heat and humidity fluxes. Setting model.surface_condition = nothing or model.boundar_layer = nothing will therefore disable all other surface fluxes unless those are have use_boundar_layer_drag = false which lets them use independently their own drag coefficient.

with more explanation below. The surface heat fluxes currently implemented are

subtypes(SpeedyWeather.AbstractSurfaceHeatFlux)

5-element Vector{Any}:
 PrescribedLandHeatFlux
 PrescribedOceanHeatFlux
 SurfaceHeatFlux
 SurfaceLandHeatFlux
 SurfaceOceanHeatFlux

and the surface humidity fluxes (this does not include Convection or Large-scale condensation those are precipitation) implemented are

subtypes(SpeedyWeather.AbstractSurfaceHumidityFlux)

5-element Vector{Any}:
 PrescribedLandHumidityFlux
 PrescribedOceanHumidityFlux
 SurfaceHumidityFlux
 SurfaceLandHumidityFlux
 SurfaceOceanHumidityFlux

The calculation of thermodynamic quantities at the surface (air density, temperature, humidity) are handled by

subtypes(SpeedyWeather.AbstractSurfaceCondition)

1-element Vector{Any}:
 SurfaceCondition

and the computation of drag coefficients (which is used by default for the surface fluxes above) is handled through the model.boundary_layer where currently implemented are

subtypes(SpeedyWeather.AbstractBoundaryLayer)

2-element Vector{Any}:
 BulkRichardsonDrag
 ConstantDrag

Fluxes to tendencies

In SpeedyWeather.jl, parameterizations can be defined either in terms of tendencies for a given layer or as fluxes between two layers including the surface flux and a top-of-the-atmosphere flux. The upward flux out of layer   into layer (vertical indexing increases downwards) is (  for half, as the flux sits on the cell face, the half-layer in between and  ) and similarly is the downward flux. For clarity we may define fluxes as either upward or downward depending on the process although an upward flux can always be regarded as a negative downward flux. The absorbed flux in layer is

A quick overview of the units

Quantity	Variable	Unit	Flux unit
Momentum	Velocity
Heat	Temperature
Moisture	Specific humidity

The time-stepping in SpeedyWeather.jl (see Time integration) eventually requires tendencies which are calculated from the absorbed fluxes of momentum or and moisture as

with gravity and layer-thickness (see Sigma coordinates) so that the right-hand side divides the absorbed flux by the mass of layer (per unit area). Tendencies for equivalently with their respective absorbed fluxes.

The temperature tendency is calculated from the absorbed heat flux as

with heat capacity of air at constant pressure with a typical value of . Because we define the heat flux as having units of the conversion includes the division by the heat capacity to convert to a rate of temperature change.

Bulk Richardson-based drag coefficient

All surface fluxes depend on a dimensionless drag coefficient which we calculate as a function of the bulk Richardson number following Frierson, et al. 2006 [^Frierson2006] with some simplification as outlined below. We use the same drag coefficient for momentum, heat and moisture fluxes. The bulk Richardson number at the lowermost model layer   of height is

with   the Geopotential at  ,    the virtual dry static energy and the Virtual temperature. Then the drag coefficient follows as

with   the von Kármán constant, the roughness length (we currently use different values over ocean and land). There is a maximum drag for negative bulk Richardson numbers but the drag becomes 0 for bulk Richardson numbers being larger than a critical   with a smooth transition in between. The height of the -th model layer is   with the geopotential

which depends on the temperature of that layer. This model layer height above the surface is recalculated on every time step for every grid cell as it depends on temperature. The maximum drag coefficient then follows as

to simplify the drag coefficient calculation to

with   the clamped which is at least and at most .

Surface momentum fluxes

The surface momentum flux is calculated from the surface wind velocities

meaning it is scaled down by   (Fortran SPEEDY default, [^SPEEDY]) from the lowermost layer wind velocities . A wind speed scale accounting for gustiness with   (Fortran SPEEDY default, [^SPEEDY]) is then defined as

such that for low wind speeds the fluxes are somewhat higher because of unresolved winds on smaller time and length scales. The momentum flux is then

with   the surface air density calculated from surface pressure and lowermost layer temperature . Better would be to extrapolate to a surface air temperature assuming adiabatic descent but that is currently not implemented.

Surface heat fluxes

The surface heat flux is proportional to the difference of the surface air temperature and the land or ocean skin temperature . We currently just approximate as the lowermost layer temperature although an adiabatic descent from pressure to surface pressure would be more accurate. We also use land and sea surface temperature to approximate although future improvements should account for faster radiative effects on compared to sea and land surface temperatures determined by a higher heat capacity of the relevant land surface layer or the mixed layer in the ocean. We then compute

in , positive up.

Surface humidity fluxes

The surface humidity flux, i.e. evaporation/condensation of soil moisture over land and evaporation of sea water over the ocean or condensation of humidity on the ocean surface is proportional to the difference of the surface specific humidity and the saturation specific humidity given and surface pressure . This assumes that a very thin layer of air just above the ocean is saturated but over land this assumption is less well justified as it should be a function of the soil moisture and how much of that is available to evaporate given vegetation, hence we include a soil moisture availability. We again make the simplification that  , i.e. specific humidity of the surface is the same as in the lowermost atmospheric layer above.

The surface humidity flux in , positive up, is then

with the saturation specific humidity calculated from the skin temperature and surface pressure . The availability of soil water over land is represented by (over the ocean  )

following the Fortran SPEEDY documentation[^SPEEDY] which follows Viterbo and Beljiars 1995 [^Viterbo95]. The variables (or spatially prescribed arrays) are water content in the top soil layer and the root layer below using the vegetation fraction    composed of a (dimensionless) high and low vegetation cover per grid cell . The constants are depth of top soil layer  , depth of root layer   (defaults subject to change, check model.land.geometry), soil wetness at field capacity (volume fraction)  , and soil wetness at wilting point (volume fraction)  .

Land-sea mask

SpeedyWeather uses a fractional land-sea mask, i.e. for every grid-point

• 1 indicates land

• 0 indicates ocean

• a value in between indicates a grid-cell partially covered by ocean and land

The land-sea mask determines solely how to weight the surface fluxes coming from land or from the ocean. For the sensible heat fluxes this uses land and sea surface temperatures and weights the respective fluxes proportional to the fractional mask. Similar for humidity fluxes. You can therefore define an ocean on top of a mountain, or a land without heat fluxes when the land-surface temperature is not defined, i.e. NaN. Let be the fluxes coming from land and sea, respectively. Then the land-sea mask   weights them as follows for the total flux

but   if the sea flux is NaN (because the ocean temperature is not defined) and   if the land flux is NaN (because the land temperature or soil moisture is not defined, for sensible heat fluxes or evaporation), and   if both fluxes are NaN.

Setting the land-sea mask to ocean therefore will disable any fluxes that may come from land, and vice versa. However, with an ocean-everywhere land-sea mask you must also define sea surface temperatures everywhere, otherwise the fluxes in those regions will be zero.

For more details see The land-sea mask implementation section.

References

[^Frierson2006]: Frierson, D. M. W., I. M. Held, and P. Zurita-Gotor, 2006: A Gray-Radiation Aquaplanet Moist GCM. Part I: Static Stability and Eddy Scale. J. Atmos. Sci., 63, 2548-2566. DOI: 10.1175/JAS3753.1.

[^SPEEDY]: Franco Molteni and Fred Kucharski, 20??. Description of the ICTP AGCM (SPEEDY) Version 41. https://users.ictp.it/~kucharsk/speedy_description/km_ver41_appendixA.pdf

[^Viterbo95]: Viterbo, P., and A. C. M. Beljaars, 1995: An Improved Land Surface Parameterization Scheme in the ECMWF Model and Its Validation. J. Climate, 8, 2716-2748, DOI:10.1175/1520-0442(1995)008<2716:AILSPS>2.0.CO;2.