VO 3+4 Parameterization I

What is not parameterization?

the dynamical core

(it solves numerically the fundamental/primitive equations of fluid dynamics)

In wich aspects can model physics be divided in?

1. Processes with fluid dynamic description, but that cannot be resolved by the model grid (zb turbulence) → subgrid processes

2. Processes with no fluid dynamic description (zb radiation)

What is dynamics?

abbreviation for “computational fluid dynamics”

What is Parameterization?

Parameterization is to express something in terms of parameters. In mathematics, it is the process of finding parametric equations for a curve, surface, or manifold.

What is model physics?

makes no sense, but has become an established term

Which sub grid processes need to be parameterized?

Processes that occur on scales smaller than the model grid must be parameterized:

• turbulence

• convection (cumulus)

• deep convection (cumulonimbus)

• microphysics

Resolution vs. effective resolution for local NWP, global NWP and Climate

Local: dx = 1-3 km → 10 km

Global: dx = 10-30 km → 100 km

Climate: dx = 50-150 km → 500 km

What can local, global and climate NWPs resolve?

Local:

• turbulence (within and above BL)

• shallow convection, both dry and moist (cumulus)

Global:

• deep convection (cumulonimbus)

Climate:

• mesoscale convective systems (they’re not parameterized separately from deep convection)

  

How many grid spacings are approximately required to a) obtain first useful information and b) fully resolve a wave?

a) about 4*dx

b) about 8*dx

Which non-subgrid processes have to be parameterized?

• ice, aerosols and their interactions with water

• radiation

• plants and other surface processes

What are the key parameterizations of ECMWF?

• clouds

• deep and shallow convection

• radiation

• turbulent diffusion

• surface heat fluxes

• orographic and non orographic wave drag

• chemistry

• ocean processes

Do all parameterizations follow universal principles or laws?

No. Every process can be parameterized in a lot of different of ways

What are common approaches for parameterizations?

• 1D column approximation

• Scale separation

Explain 1D column approximation

based on the assumption that the horizontal effects of processes can be neglected

→ simplifies problem conceptually and computationally

Explain Scale separation

assumption that the spatial and temporal scale of a parameterized process is significantly smaller than the model grid spacing and time step

→ so we parameterize the mean net effect of the process on the grid box

What are the core variables of NWP models and how are they summarized?

T, u, v, w, q, p

• temperature

• 3D wind

• humidity

• pressure

summarized as model state X

On what does the dynamic tendency depend?

only on the core variables

What is one of the main tasks of parameterization?

to quantitatively estimate the net effect of a process on the core variables

What is the simplest case of parameterization?

Tendency X = dynamic tendency (X) + parameterized tendency (X)

X_(n+1) = X_n + Tendency X *delta t

Why is physics–dynamics coupling complicated in real models?

Because models use many interacting parameterizations with additional prognostic variables (zb. cloud water, TKE, convective updrafts)

The parameterizations may:

• depend on other prognostic variables

• parameterized tendencies depend on other variables or tendencies of core/other variables

• define and calculate the same quantities differently

• use different grids and time steps than the dynamics

• must also be coupled to the land surface and ocean

Why can core variables also require parameterization? Give example

When their value is needed in between grid points

example: 2m Temperature & 2m Humidity

What happens in sequential splitting?

Dynamics and physics are calculated one after another, the updated state from one process is passed to the next

What happens in parallel splitting?

Dynamics and physics calculate their tendencies independently from the same initial model state

Main characteristics for sequential splitting?

• conservation is ensured

• the order of parameterization has a significant effect → swapping order can lead to significantly different climate sensitivity within the same model

• recommended to compute the fastest processes first

• numerically less effective (waiting for the preceding process)

Main characteristics of parallel splitting?

• conservation issues , zb mass conservation

• numerically more effective

• requires a shorter time step to remain stable

Lecture example liquid cloud fraction definition:

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

• cloud cover is similar but over entire column (2D field)

• liquid clouds are much simpler than ice, cloud droplets are commonly in thermal equilibrium

• by assuming equilibrium → cloud fraction (almost) a sub grid process

→ meaning its value at a given model level is determined by the statistical distribution of moisture within the grid box rather than by a time-evolving prognostic equation

What is an advantage of sub grid parameterizations?

Higher-resolution simulations can be used as reference truth

How is a LES (Large Eddy Simulation) used as a reference truth for a parameterization?

• it’s split into many individual slices, the statistical properties of the variables in the slice are used as input for parameterization

• parameterized cloud fraction (cf) can be directly compared to cf of LES slice

  

Types of parameterization for cf? (4)

• all or nothing (used in LES)

• relative humidity from Sundqvist

• Beta PDF without skew (symmetrical)

• Beta PDF with variable skew

All or nothing assumptions, input variables, calculation, tuning parameters?

Assumptions:

1. there is no subgrid-scale variability in moisture in the grid box (every point same moisture as box mean)

2. the grid box is in thermal equilibrium

Input variables:

• temperature

• pressure

• total moisture (= water vapor + cloud water {ohne ice})

Calculation:

1. calculate saturation water content from temperature and pressure

2. if total moisture >= saturation content

   1. then cloud fraction = 1 (100% bewölkt)

   2. else cloud fraction = 0 (0% bewölkt)

Tuning parameters: 0

Relative humidity from Sundqvist assumptions, input variables, calculation, tuning parameters?

Assumptions:

1. cloud fraction is a function of relative humidity (eta) and is 0 below a critical value

2. wenn rh = 1, cloud cover = 1

Input variables:

• temperature

• pressure

• total moisture (= water vapor + cloud water {ohne ice})

• surface pressure

Calculation:

$$cf_{rel}=1-\sqrt{\frac{1-\eta}{1-\eta_{crit}}}$$

Tuning parameters: 3

• in ICON there is a fourth parameter over the ocean (siehe Bild)

• → C2 is critical value at surface, C1 and C3 determine how value eta_crit changes with height

What is total water PDF parameterization?

replaces all or nothing with more physical approach

Assumption:

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

• there is a uniform saturation value throughout the grid cell (r_s=f(p_quer,T_quer))

$$cf=\int_{r_{s}}^{b}PDF\left(r_{t}\right)dr_{t}$$

Beta distribution

• 4 degrees of freedom: min and max value (a,b), two shape parameters (p>0, p<0)

• if p and q > 1 , it starts and ends at 0

• advantage: doesn't go to infinity

• symmetric when p=q

• complicated formula

  

Beta PDF symmetric closures, input variables, calculation, tuning parameters?

Two closures → two remaining degrees of freedom:

• from (q-1)(p-1)=2 and p=q, p and q are given

• only a and b remain undetermined

Input variables:

• temperature

• pressure

• water vapor

• cloud water

Calculation.

• saturation value calculated from T and p

• a,b determined iteratively

• cloud fraction calculated from saturated area of distribution

Tuning parameters: 0

Beta PDF with skewness closures, input variables, calculation, tuning parameters?

One closure, 3 remaining degrees of freedom:

• (q-1)(p-1)=2

Input variables:

• temperature

• pressure

• water vapor

• cloud water

• total water skewness

Calculation.

• saturation value calculated from T and p

• a,b determined iteratively

• cloud fraction calculated from saturated area of distribution

Tuning parameters: 0

Meaning of positive skewness

• means there are small regions with very high total moisture

• caused by convection

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

Meaning of negative skewness

• means there are small regions with very low moisture

• less common

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

Meaning of no skewness

• means that the distribution is symmetric, high values as common as low values

What variables besides pressure, temperature, and humidity are needed, and how many tuning parameters exist?

• "All or nothing": No additional variables

• Relative humidity from Sundqvist: No additional variables, 3 tuning parameters

• Beta PDF without skewness (Symmetric): Cloud water

• Beta PDF with variable skewness: Cloud water, skewness of the total water PDF

What is the most difficult aspect of Beta PDF?

• they require additional variables

• these variables can be obtained from LES in offline test, but in a model they must be provided by other parameterizations

• cloud water would be provided by the microphysics parameterization

• skewness is deduced by convection and BL parameterizations

What is this?

cloud fraction in the LES slices

y-axis: LES cloud fraction

x-axis: total water vs. saturation

Evaluation of the parameterizations against the LES cloud fraction:

• Y-axis: Error in the parameterized cloud fraction

• Errors are largest at relative humidity near 1

• "All or nothing" performs poorly.

• Relative humidity needs to be retuned for

  each grid size. The CMIP5 values had a very small mean error for 100x100 km boxes, but a relative absolute error of 100% (GSN2018)

• The Beta PDF approach has much smaller errors, and the variable skewness provides an additional improvement

Code example: diagnostic cloud fraction, stratiform liquid cloud

$$cc_{turb}=\left(\frac{q_{v}+q_{c}+A\Delta q-q_{sat}}{B\Delta q}\right)^2$$

delta q: variance of the total-water PDF, determined by the turbulence scheme

A: tunable parameter, determines asymmetry of total water distribution

B: tunable parameter, scales the cloud cover for a determined PDF asymmetry (more or less clouds with same critical relative humidity, B=A+1 im code)

Why so much code for something so simple?

• lots of limiting things within safety margins

• tuning parameters A and B adjusted on height

• delta q adjusted for layer thickness

• specific tweaking to enhance cloud fraction in stratocumulus region

• adjusts cloud liquid water under specific conditions