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Horizontal Mixing and Convection
• 1st order prognostic PDE
• Q is the quadratic fluid
dynamics term called the
“Dynamical Core”.
• F is the Forcing “Physics”
and dissipation.
Forcing Terms:
• radiation
• clouds
• convection
• horizontal mixing/transport
• vertical mixing/transport
• soil moisture and land
surface processes
• greenhouse gases
• marine biogeochemistry
• volcanoes
External Gravity Wave
Subgrid Scale modeling
Subgrid Scale modeling of
a Rossby Wave.
Reynold’s Averaging:
•
Averaging over Depth
(weight by density):
Useful relations:
= mean + eddy
•Averaging over both latitude and
longitude:
Reynold’s Averaged Numerical Simulation (RANS)
Average the equation over grid boxes:
Parameterize the mean vorticity flux due to subgird size eddies:
Or more correctly model the flux of unresolved scales as function
of the mean plus random noise => Stochastic Physics.
Closure:
In order to solve the problem,
assumptions about the unknown
quantities are needed.
Applied to our vorticity flux:
K-theory: connects the
fluxes of the trace species to the
gradient of the mean quantities through the
eddy diffusivity i.e.
In this example, the kinematic flux of a
pollutant is modeled as being equal to
the eddy thermal diffusivity, K, times the
vertical gradient. The assumption is that
the concentration always flows
down-gradient.
Note: Mixing is not always
down-gradient for example:
vorticity fluxes due to midlatitude storms act to
strengthen the westerly jets.
Smagorinsky-Lilly Model
suggests that the viscosity
be a combination of the
molecular viscosity and the
sub-grid-scale viscosity.
Buildup of energy at the
smallest scales of the
model.
Energy from resolved scales
needs to be transported to
unresolved scales.
Richardson’s parody of
Jonathan Swift’s poem:
Greater whirls have lesser whirls
That feed on their velocity
And lesser whirls have smaller whirls
And so on to viscosity
The precipitation from April, 2005 to April, 2006
displayed in this loop is computed in the
NASA/GSFC Laboratory for Atmospheres as a
contribution to the GPCP, an international research
project of the World Meteorological Organization's
Global Energy and Water Exchange program.
Detailed information and a variety of precipitation
products and images are available at
http://precip.gsfc.nasa.gov/ .
Precipitation is caused by rising of air due to:
 Orographic Uplift
 Convective Instability -- Rising Thermals (Local)
 Frontal Lifting
 midlatitude cyclones form along the polar front.
 near the equator where the trade winds meet at
the ITCZ.
Reference: K. Emmanuel, 1994: Atmospheric Convection, University of Oxford
Press. Chapter 16.
Convective Parameterization
Temperature tendency = adiabatic large
scale forcing + diabatic heating from
condensation of water (due to cumulus
convection). Q1 can also include radiative
terms.
q is the specific humidity.
net water vapor mixing radio tendence =
large scale advection + condensation (Q2 is
in terms of the latent heat so that it can be
directly compared to Q1).
Closure Conditions:
• dT/dt connected to dq/dt => constrain the large scale average states and forcing
• Q1 and Q1 => constrain the moist convective processes
• Q1 and/or Q2 with dT/dt and/or dq/dt => constrain the coupling between large and
small scales.
Adjust the temperature and
moisture lapse-rates to the
reference profiles.
 Convection occurs when
1. Model layer is saturated
(RH>100%)
2. Layer is conditionally unstable
>m
Usefulness: Computationally
cheap and easy.
Does the right thing in broad
Strokes => rains in the
tropics.
Problem: No Clouds.
No downdrafts.
Instantaneous.
Unrealistically high rainfall.
Soft Convective Adjustment
Relax the constraint on 100% humidity.
Assume convection only occurs over a fraction of the grid.
Good points:
More realistic
Gradual relaxation -- less instantaneous (sometimes)
Still computationally cheap
Bad points:
Still constrained by reference profiles
More tunable parameters
No explicit spatial or time correlation
Moisture Convergence
into a model column,
M.
e is the surface
evaporation.
Integrate from the surface
to the tropopause.
A fraction of the moisture
remains in the
atmosphere and the
rest rains out.
s is the dry static
energy. The vertical
heat and moisture flux
is proportional to the
difference between
the cloud values and
the averaged state.
Balance equations or simpler 1D cloud
models are used for closure.
Mass Flux Model
• Assumes that the collective behaviour of a group of
clouds can be represented as a bulk cumulus cloud.
• Good points:
gives more realistic rainfall rates.
easily generalized to include cloud dynamics.
based on physics (?).
• Bad points:
convection isn’t driven by mass flux but by local
instability.
many more parameters to approximate.
computationally expensive.
Model Output
/home/sws00rsp/noise/km12.html
•
Independent Calculations for each grid box. There’s no
communication between boxes.
•
There’s no memory to the calculations.
•
Need equilibrium (instantaneous) assumptions to get closure.
• Adjustment Schemes work well for coarse resolution and they
are cheaper and faster.
•
Forecast models run at high enough resolution (1-1.5km) that
they don’t need parameterizations…
Parameterizing Convection
•
Trigger function for each box
– Bouyancy of the parcel (will it ascend or not?)
•
Vertical extent of Convection
– Amount of Entrainment
•
or du/dt changes in large-scale flow.
Closure
– % of convective activity
– Scale factor
100 m
LEM
Model Scale
1.5km
4km
12 km
MET OFFICE MODELS
Better without
Weakened
Scheme
scheme
20-30km
Complex Schemes
add physics to
model
convective storms.
100 km
Simple Adjustment
Move back to the
Moist adiabat
It rains in the
tropics.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.