VO4 - Parameterizations 2

What is liquid cloud fraction?

fraction of a model grid cell that is covered by water clouds, defined in each vertical grid cell (3D field)

What is cloud cover?

similar to liquid cloud fraction but over the entire column (2D field)

What is a advantage of subgrid parameterizations?

higher-resolution simulations can be used as reference truth.

What is LES?

Large Eddy Simulations - models with extremly high resolution on a confined region

dx ~150m

What does this plot show?

Satellite image vs LES with 150m grid resolution

LES resolution still to coarse to resolve small clouds but thats ok as it should only give a possibility thats realistic

Explain the approach with LES as reference truth.

• LES is split up into many individual slices. The statistical properties of the variables in the slice are used as input for the parameterization

• parameterized cloud fraction can be directly compared to cloud fraction of the LES slice

What is Total moisture?

Total moisture = water vapor + cloud water

(ice ignored)

What does this plot show?

Histogram of total moisture / saturation moisture (=RH) against cloud fraction

The larger the grid box, the less likely it is to reach a cloud cover of 1

even for 3.5km a cloud fraction of 1 is quite rare

even for RH~30% clouds pop up in the 55km slices

Which 4 parameterizations for liquid cloud fraction are there?

• All or nothing

• Relative humidity from Sundqvist

• Beta PDF without Skew

• Beta PDF with variable Skew

What are assumptions in the “all or nothing” parameterization?

• no subgrid-scale variability in the grid box
  (works well for small grid box and liquid water)

• grid box in thermal equlibrium

What are input variables in the “all or nothing” parameterization?

• Temperature

• pressure

• total moisture

How does the calculation in the “all or nothing” parameterization work?

• Calculate saturation water content from the core variables T and p

• if total moisture >= saturation water content → cloud fraction = 1

• if not → cloud fraction = 0

How many tuning parameter are there in the “all or nothing” parameterization?

0

What assumptions are made in the Relative humidity from Sundqvist?

• Cloud fraction a function of relative humidity and is 0 below a critical value

• at RH = 1 = 100%, cloud fraction = 1

What are the input variables for the Relative humidity from Sundqvist?

• Temperature

• total moisture

• pressure

• surface pressure

How does the calculation of the Relative humidity from Sundqvist look?

3 Tuning parameters

What assumptions are made in the Total water PDF parameterization?

• distribution of total moisture in the grid cell is described by an analytical distribution

• there is a uniform saturation value throughout the grid cell, which is a function of the mean pressure and the temperature:

How does the calculation for the Total water PDF parameterization?

r_s from the second assumption

How can the integrals in the Total water PDF distribution be understood?

integration over pdf between r_s and b gives cloud fraction c_f

if we also take r_t - r_s into account, this gives us the liquid water amount r_c

How many degrees of freedom does a Beta distribution have?

4 doF

• min, max of (a, b)

• two shape parameters p,q > 0

expl Gaussian would have 2 doF

When does the Beta distribution start and end at 0?

If p and q > 1

What is the major advantage / difference of the Beta distribution compared to the normal distribution?

does not go to infinity

When is the Beta distribution symmetric?

p = q

Which two closures can be used to reduce the doF of the Beta distribution?

• (q - 1)(p-1) = 2, with p,q > 1
  yields 3 doF
  using only this closure yields a Beta PDF with flexible skewness

• q = p
  yields 2 doF
  using this closure as well yields a Beta PDF symmetric

What are the 2 doF of the Beta PDF symmetric?

from (q-1)(p-1)=2 and q=p → q, p known

only a and b remain undetermined

What are the input variables of the Beta PDF symmetric?

• Temperature

• pressure

• water vapor

• cloud water (not from the dynamical core!)

How does the calculation for the Beta PDF symmetric look?

• Saturation value is calculated from T and p

• a, b determined iteratively, so that the integrals for total moisture r_t and cloud water r_c correspond to the input variables

• cloud fraction calculated from saturated area of distribution

How many tuning parameters does the Beta PDF symmetric have?

0

What are the 3 doF of the Beta PDF with flexible skewness?

• from (q-1)(p-1) = 2 still only one combination for a certain skewness

• therefore a, b, and skewness are the doF

What are the input variables for the Beta PDF with flexible skewness?

• Temperature

• pressure

• water vapor

• cloud water (not from dynamical core!)

• total water skewness (not from dynamical core!)

How does the calculation for the Beta PDF with flexible skewness look?

• Saturation value r_s calculated from given T and p

• thanks to closure, p, q determined from skewness

• the free parameters a, b are iteratively determined, so that the integrals for r_t and r_c correspond to the input variables

• cf corresponds to the saturated area of the distribution

How man tuning parameters does the Beta PDF with flexible skewness have?

0

What does positive skewness mean?

• there are small regions with very high total moisture

• caused by convection

• leads to smaller cloud fraction for a given amount of cloud water

What does no skewness mean?

• distribution is symmetric

• high values as common as low values

What does negative skewness mean?

• small regions with very low moisture

• less common

• e.g. Sc-fields

• leads to higher clouds fraction for a give amount of cloud water

What are the so called core variables?

T, p, humidity

What is the most difficult aspect of the Beta PDF parameterization?

additional variables are required

Where can the additional variables for the Beta PDF parameterization be obtained?

either LES in an offline test or from other parameterizations in a model

From which parameterization can the cloud water be obtained?

from the microphysics parameterization

From which parameterization can the skewness be obtained?

convection and BL parameterization

How does convection impact skewness?

increases skewness

How does turbulence / mixing impact skewness?

decreases skewness

What does this plot show?

• y—axis: error in parameterized cloud fraction

• errors largest for RH~1

• “all or nothing” performs poorly, only ok below RH=1.2 and above RH=0.6

• Beta PDF approach much smaller errors and variable skewness provides additional improvement