Sub grid scale transport by turbulent diffusion:
a consequence from the numerical treatment of principal model equations
Matthias Raschendorfer
DWD
Matthias Raschendorfer
CLM-Training Course 2011
The model equations:
• Consider the following at least 5 prognostic variables:
φ ∈ M : = {v i ; i = 1,2,3}
= {u , v , w }
φ ∈ H : = {q w ,
θw }
velocity components (momentum
concentration)
momentum
mass fraction of water phases in the
atmosphere
a temperature variable related to
thermal energy
scalars
• Prognostic budget equations:
∂ t (ρφ) +
∇ ⋅ Fφ = Qφ
Fjφ = ρ φ v j − c φj
ρ
DWD
general budget
j-component of the total flux density for the property φ
air density
Matthias Raschendorfer
CLM-Training Course
• with the non advective molecular flux density components:
c φj
− a φ ∂ j φ
=
− µ ∂ i v j + ∂ j v i
(
)
for scalars
φ = θw , qw
for momentum φ = v i
a φ = ρ k φ : molecular austausch coeff.
(i,j)-component of the reduced
molecular stress tensor
a v i = µ = ρ υ : isotropic dynamic viscosity
• and the source terms:
L
 ∇⋅S
≈ 0 (for moist adiab. idealization) , φ = θ w = θ − c q c
−
π c pd
c pd

Q φ = 0
,φ = q w = q v + q c
− δ ρ g − 2ρ(Ω × v) − ∂ p
,φ = v i
i
i
 i3

(conservative)
total water potential temperature
conservative
total water mass fraction
momentum component in i-direction
not conservative
may also be regarded as a part of the i-th
diagonal element of molecular stress tensor
S : radiation flux density
Ω : angular vector of the earth
g : gravity of the earth
 p
π = 
 pr
Rd
 c pd

: Exner factor

p r = 1000 hPa : reference pressure
c p d : heat capacity of dry air at constant pressure
c p v : heat capacity of w ater v apor
R d : specific gas cons tan t of dry air
R v : specific gas cons tan t of water vapor
q v : m ass fraction of water vapor (specific humidity
DWD
θ = T / π : potential temperat.
)
q c : m ass fraction of cloud water
Matthias Raschendorfer
CLM-Training Course
• together with some diagnostic relations:
 R


p = ρR d 1 +  v − 1q v − q c  T
  R d 

 q 
 qv 
  = sat _ adj  w  ,
 θ w 
T
DWD

p

equation of the state (gas law)
saturation adjustment
Matthias Raschendorfer
CLM-Training Course
The coordinate system:
• The equations must be solved on a numerical grid.
-
For our purpose we choose a regular grid, belonging to a local Cartesian
coordinate system (x, y , z ) with a physical vertical coordinate σ ,
such that
z = z σ (x, y )
may be terrain following
and
∂σz = 1
within the boundary layer.
z
zσ
σ = const
∆z = ∆σ
∆x
DWD
∂j ζ = ∂j ζ σ − ∂jz σ ∂zσ∂σζ σ , j ∈{1, 2} =ˆ { x, y}
∂3 ζ = ∂zσ ∂σζ σ ,
3 =ˆ z
x
Matthias Raschendorfer
CLM-Training Course
The filter operator:
• In order to apply a numerical approximation of the derivatives in space on the variable fields, they have to be
filtered in space with respect to a horizontal length scale comparable to the horizontal grid spacing
D
g
= ∆x = ∆y .
• The filter is defined as a moving average along the air containing part
in the
ς(r) :=
z
σ-system (marked with an over-bar):
ς (s) ds3
∫
(r) s∈G (r)
1
Gσ
⋅
σ
volume average
ςˆ : =
σ
ρς
ρ
G σ (r )
of a shallow cubic grid box
weighted average
φ
∂Q
G n
E
∂ xφ
∆φ / ∆x
∂ x φ ≈ ∆φ / ∆x
y
σ
Q σ (r )
∆x
Q
r
∆y
x
x
ζ′ (s) := ζ(s) − ζ(r )
DWD
sub grid scale
deviation
ζ′′ (s) := ζ(s) − ζˆ (r )
Matthias Raschendorfer
∆x = D g
CLM-Training Course
The concept of ”topographic approximation”:
Sub grid scale slope of the
model layers or non
atmospheric inclusions
Roughness terms in the
averaged model equations
Averaging and
differentiation in space (flux
divergence) can no longer
be commuted
___
( )′
generating slope
′
correlation terms: ∂ j φ = ∂ j φ + φ ∂ j z
with respect to z σ
Wake turbulence
production for TKE
σ
slope correlation terms in
case of sub grid scale slopes
• For any real surface of the earth, there exists an “equivalent topographic surface” ET,
which has the same surface area as the real surface.
z
z
σ = const
0
DWD
CLM-Training Course
The filtered gradient terms in topographic approximation:
( ) ′ = 0 for all scalar variables φ ≠ p (not for the pressure)
′
• We assume φ ∂ j z σ
and for v i ≠ j
σ
• But the correlation appears in:
v̂ j
f jφ
The averaged flux divergence term:
eφ
Using
f φ : = ρφ′′v′′ − a φ∇φ
molecular gradient flux density
∇ ⋅  ρ φ ′′~v ′′ + ~e φ 


it is for scalars:
φ
( )
∇ ⋅ F = ∇ ⋅ ρ φˆ ˆv
φ
+ ∇ h ⋅ fh
grid scale
divergence
of the grid
scale flux
DWD
p lee < p luv
(local flux correction)
σ

+ ∂ z  − fh φ ⋅ ∇ h z

σ
∆x j
(
− fh φ ⋅ ∇ h z
σ
)′
sub grid scale
sub grid scale
horizontal
 ∂ x   ∂ x − ∂ x z ∂ z 

horizontal
flux


flux
∂z 
divergence
of
 = ∇ − ∇ h zhorizontal
∇ =  ∂ y  =  ∂ y through
− ∂ y z ∂ z  theh
through
the


the sub grid
 

∂ z   ∂ grid scale σ
 σ
sub grid scale
scale flux along
surface
σ- surface
a σ -surface
σ
σ
σ
σ
σ
Matthias Raschendorfer
σ
xj

+ f zφ 

sub grid
scale
vertical
flux
CLM-Training Course
The closure problem:
• Due to nonlinearity we get second order statistical moments being additional model variables:
e.g., we obtain for the non linear flux term:
ρφ v j = ρ φˆ vˆ j + ρφ′′v ′j′
• Further information about these additional variables has to be introduced:
- we call this “parameterization of sub grid scale processes”
- and distinguish different processes more or less related to the length scale of motions
- each with specific parameterization assumptions
Turbulence:
isotropic, only one characteristic length scale at each grid point
Circulation:
non isotropic, coherent structures of additional length scales,
e.g. convection
DWD
Matthias Raschendorfer
CLM-Training Course
• 1-st order closure:
- Parameterizations for the statistical moments are introduced ad hoc in the 1-st order budgets
- A mass flux scheme is a sophisticated 1-st order closure for convection
• Higher order closure for turbulence:
- prognostic equations for the unknown statistical moments of order 2 (in general n) are considered
- They contain additional statistical moments up to order n+1.
Closure dilemma of hydrodynamics
- We use a 2-nd order closure model according to Mellor/Yamada for turbulence.
statistical moments in the 2-nd order budgets have to be parameterized
DWD
Matthias Raschendorfer
CLM-Training Course
General second order budget equation:
(including roughness layer terms in topographic approximation)
(
) (
)
shear production
sub grid scale macroscopic transport
(
ˆ  − ρψ′′~v′′ ⋅ ∇φˆ + ρφ′′~v′′ ⋅ ∇ψ
ˆ
Dt ρφ′′ψ′′ := ∂ t ρφ′′ψ′′ + ∇ ⋅  ρφ′′ψ′′ ~ˆv + ρφ′′ψ′′~v′′ + φ′′~eψ + ψ′′~e φ  = − ~e ψ ⋅ ∇φˆ + ~e φ ⋅ ∇ψ

 

molecular dissipation
+  eψ ⋅ ∇φ′′ + e φ ⋅ ∇ψ′′ 
neglected
outside
the
φ
φ


molecular flux density e : = −a ∇φ
laminar layer
+  φ′′Qψ + ψ′′Qφ 


vanishing for conservative variables
)
source term correlation
− φ′′∂ ip , ψ = v i
− ∂ i φ′′p′
δ i3
g
ρφ′′θ′v′ ≈ − φ′′ ∂ i p
ˆθ
v
+ p′∂ i φ′′
(return-to-isotropy)
DWD
pressure transport
≈0
( )
+ p′(∂ z )′ ∂ φˆ
+ φ′ ∂ i z σ ′ ∂ z p
i
σ
z
buoyancy source
pressure correlation
(wake production)
Matthias Raschendorfer
CLM-Training Course
Postulates of pure turbulence:
• Equilibrium of the source terms in all 2-nd order budgets
(if the turbulent stress tensor is substituted by its traceless form) :
- neglect of local time derivative
- neglect of (grid scale and sub grid scale) transport
- neglect of correlations with source terms of 1-st order budget equations
• Spectral density of 2nd-order moments follows a power law in terms of wave length in each direction
(inertial sub range spectrum):
- whole SGS spectrum in a given sampling direction is determined by a single peak wave length L p
• Peak wave length is the same for samples in all directions: isotropic length scale
- pressure correlation and dissipation can be closed using a single turbulent master length scale
for each location according to Rotta and Kolmogorov
l
Turbulence is that class of SGS structures being in agreement with turbulence closure assumptions!
DWD
Matthias Raschendorfer
CLM-Training Course
Single column solution for turbulent flux densities:
1.
Using closure assumptions valid for pure turbulence:
2-nd order budgets reduce to a 15X15 linear system of equations
built of all second order moments of the variable set { θ w , q w , u, v , w }
2.
Using general boundary layer approximation:
- neglect derivatives of mean quantities along filtered topographic surfaces
compared to derivatives normal to that surfaces
Flux gradient representation of the only relevant vertical flux densities:
General boundary layer approximation:
turbulent master length scale
ρφ′′w′′ ≈ ρ φ′w′ ≈ −ρKφ ∂ z φˆ
K φ : = Γ l S φ ⋅ q turbulent diffusion coefficient
• iso surface with an area being
Γ times the horizontal area
1 3
2
turbulent velocity scale  ∑ ρv ′i′ 
 ρ i=1

squared surface area index
stability function
(only inside the roughness layer)
2 ⋅ TKE
Γ and
l
2
roughness
layer
earth
are both functions of the location only.
S φ and q are functions of Γ ,
DWD
1
l , vertical wind shear and buoyancy (thermal stratification).
Matthias Raschendorfer
CLM-Training Course
The moist extension:
• Inclusion of sub grid scale condensation achieved by:
- Using conservative variables with respect to condensation: q w = q c + q v
θw = θ −
Lc
qc
c pd
Correlations with condensation source terms are considered implicitly for non precipitating clouds.
q v and cloud fraction rc
cloud water q c
• Solving for water vapor
by using the statistical saturation adjustment scheme
(according to Sommeria/Deardorff):
-Normal distribution of oversaturated cloud water ∆q sat
-Expressing variance of ∆q sat by variance of θ w and q w , both generated from the turbulence scheme
∆q sat
w
cloud generated by sub
grid scale condensation
0
x
DWD
Matthias Raschendorfer
CLM-Training Course
Time-Height cross sections:
Stratification:
stable
non stable
residual
layer
non stable
mixed
layer
low level jet
mixed
layer
stable
DWD
Matthias Raschendorfer
CLM-Training Course
due to vertical shear
DWD
Matthias Raschendorfer
due to
positive
buoyancy
CLM-Training Course
Recent generalizations obtained by scale separation:
Application of closure assumptions leads to limited (not general valid ) solution:
General valid 2-nd order closure assumptions can’t exist!
General valid sub grid scale closure is possible by
-
separation of the sub grid scale flow in different classes with specific closure assumptions
-
consider of interaction terms between the different scales
averaging 2-nd order budgets with respect to the separation scale L = min L p , D g
{
}
along the whole grid cell (double averaging) leads to turbulent budgets with additional
production terms due to shear terms with respect to the separated sub gird scale
circulation flow of
DWD
•
wake vortices by SSO (sub grid scale orography) blocking
•
horizontal shear vortices
•
surface induced density flow patterns
•
shallow and deep convection patterns
Matthias Raschendorfer
CLM-Training Course
Separated semi parameterized TKE equation (neglecting molecular transport):
mean (horizontal) shear
production of circulations,
L
( )
″
of − v̂ L ⋅ ∇ p
of − v′′ ⋅ ∇ p L
1

∂t  ρ ⋅ q L 2 
2

time
tendency
=







θ̂ v
transport
buoyancy
(advection
production
+ diffusion)

according
Kolmogorov
3
L
i =1

shear production
by the mean flow

shear production
by sub grid scale
circulations
≥0
≥0
v
Matthias Raschendorfer

i =1
2
i =1
L

 

+ g ρθ′v′ w ′′ L +− ∑ ρv ′i′ ~v′′ ⋅ ∇ ˆv i  + − ∑ ρv ′i′ ~v′′ L ′ ⋅ (∇ ˆv i ) L ′  +
3
labil: > 0
neutral: = 0
stabil: < 0
DWD
− µ ∑ ∇v i
to be parameterized
by a non turbulent
approach
expressed by turbulent
flux gradient solution
 ρ q 2 ~ˆv

L
1 
∇ ⋅
2  3
2~ 

 + ∑  ρ v ′i′ v′′ 
L
 i=1
3
buoyant and wake part
buoyant part
: with respect to a
separation scale L
3

qL 
− ρ

MM

α l


eddydissipation
rate (EDR)
<0
v
CLM-Training Course
Effect of the density flow driven circulation term for stabile stratification:
σ
σ
D t (ρ TKE) ≈ −ρu′′w′′ ⋅ ∂ z uˆ +
g
ρθ′v′ w ′′ − ε
θv
turbulent
buoyancy flux
circulation
buoyancy flux
θ̂ v
ˆ
ρLw
D
σ = HC
HC
ρw ′′θ′v′
L
L
″ˆ ″
θV
L
= − ρ K ∂ z θˆ
(1 − a ) ⋅ D
a ⋅D
σ = const.
θ +v
σ=0
x
θ v−
0
0
ρθ′v′ w′′
horizontal scale
of a grid box
• Even for vanishing mean wind and negative turbulent buoyancy there remains a positive definite source term
∆x
0
TKE will not vanish
DWD
Solution even for strong stability
Matthias Raschendorfer
CLM-Training Course
Current model realization:
• in GME:
- diagnostic TKE equation (only right hand side in balance)
- without circulation production and wake production
- no surface area function considered
• in COSMO (LME, LMK):
- Prognostic TKE equation
- Implicit consideration of SGS condensation by use of statistical saturation adjustment scheme
In the operational configuration, a different SGS condensation diagnosis is used for radiation!
- Additional (crude representations of)
shear production due to sub grid scale
- horizontal shear modes,
- topographic wakes,
- surface induced density flows and
- convection (only in test version)
being together always a positive source, even in very stable situations
without mean vertical wind shear!
- More processes can be described compared to the diagnostic scheme.
- No Critical Richardson number (realisable scheme even for very stable stratification)
- In principal we can do it without a minimal diffusion coefficient.
- Development of inversions and turbulence near the jet stream or convective cells can be
described more physically.
- TKE-advection and the (vertically resolved) roughness layer are not yet implemented!
DWD
Matthias Raschendorfer
CLM-Training Course
Vertical cross section through a jet stream:
new prognostic TKE scheme:
old diagnostic TKE scheme:
no physical solution almost
everywhere above the ABL
DWD
Matthias Raschendorfer
CLM-Training Course
INPUT-parameters for the turbulence scheme:
itype_turb:
1:
type of turbulence parameterisation
former calculation of the turbulent diffusion coefficients in the atmosphere using subroutine
“parura”
3:
new turbulence scheme with prognostic TKE equation, using subroutine “turbdiff”
5_8:
different versions of a more simple Prandtl/Kolmogorov-approach introduced for comparison
imode_turb:
0:
modus for calculation of vertical turbulent flux divergences
implicit treatment of the dry part of vertical diffusion like before, using a concentration condition
at the lower boundary
1:
like 0, but with a flux condition at the lower boundary
2:
explicit treatment of vertical diffusion
3:
alternative implicit treatment of vertical diffusion based on the fluxes in conservative variables
(going to be changed in order to get rid of explicit SGS condensation corrections)
itype_sher:
type of wind shear calculation in TKE equation
1:
only turbulent vertical shear forcing of horizontal wind
2:
turbulent isotropic 3D-shear forcing of wind
3:
non isotropic 3D-sher forcing by considering a separated horizontal shear mode
DWD
Matthias Raschendorfer
CLM-Training Course
icldm_turb:
treatment of clouds with respect to turbulence
-1:
ignoring cloud water completely (pure dry scheme)
0:
no clouds considered (all cloud water is evaporated)
1:
only grid scale condensation possible
2:
sub grid scale condensation by one of the two versions of subroutine “coud_diag”
itype_wcld:
1:
type of new cloud diagnostics in subroutine “coud_diag”
diagnosis of water clouds, using subroutine “cloud_diag“ with that version based on relative
humidity (similar to the procedure of the radiation scheme but without special tuning)
2:
icldm_rad:
diagnosis of water clouds, using the statistical cloud scheme in subroutine “cloud_diag “.
treatment of clouds with respect to radiation
0:
radiation does not “see” any clouds
1:
radiation “sees” only grid scale clouds
2:
radiation “sees” clouds, being diagnosed by one of the two versions of subroutine “coud_diag”
3:
radiation “sees” clouds, being diagnosed with the former scheme but with a correction
concerning the convective cloud cover
4:
DWD
radiation “sees” clouds, being diagnosed exactly with the former scheme
Matthias Raschendorfer
CLM-Training Course
lexpcor:
switches on the above mentioned explicit correction
----------------------------------------------------------------------------------------------------------------------------------------------ltmpcor:
switches on the calculation of temperature tendencies related to conversions of inner energy
to TKE, (should be FALSE, because the effect is very small)
lnonloc:
switches on the non local option (is not tested yet and should be FALSE)
lcpfluc:
switches on the effect of fluctuating humidity on the heat capacity of air in the calculation of
the sensible heat flux (should be FALSE, because the effect is only small)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
limpltkedif: switches on the implicit calculation of TKE-Diffusion
ltkesso:
wake production of TKE due to the influence of sub grid scale orography (SSO) active
ltkecon:
TKE production due to kinetic energy transfer from sub grid scale convection active (only test
version)
DWD
Matthias Raschendorfer
CLM-Training Course
Length scale (factors) for turbulent transport:
tur_len
= 500.0
asymptotic maximal turbulent length scale [m]
pat_len
= 500.0
length scale of subscale surface patterns over land [m] (scaling the circulation term)
c_diff
= 0.20
length scale factor for vertical TKE diffusion (c_diff=0 means no diffusion of TKE)
a_hshr
= 0.2
length scale factor for separated horizontal shear mode
Dimensionless parameters used in the sub grid scale condensation scheme (statistical cloud scheme):
clc_diag
= 0.5
cloud cover at saturation
q_crit
= 4.0
critical value for normalized over-saturation (original setting q_crit=0.16)
c_scld
= 1.00
factor for liquid water flux density in sub grid scale clouds
Minimal diffusion coefficients in [m^2/s]:
tkhmin
= 1.0
for scalar (heat) transport
tkmmin
= 1.0
for momentum transport
to avoid too much low level cloud over ocean
Numerical parameters:
epsi
= 1.0E-6
relative limit of accuracy for comparison of numbers
tkesmot
= 0.15
time smoothing factor for TKE and diffusion coefficients
wichfakt
= 0.15
vertical smoothing factor for explicit diffusion tendencies
securi
= 0.85
security factor for maximal diffusion coefficients
DWD
Matthias Raschendorfer
CLM-Training Course
Thank You for attention!
An extension of this talk including more details in particular about the stability functions and the first
attempts of expressing circulations terms due to separated modes for horizontal shear, orographic
wakes and thermal direct (buoyant) circulations is existent but not included here, since it is beyond the
scope of this lecture. It is being prepared for some kind of publishing together with an conceptual treatise
of this generalized concept of turbulence modelling.
DWD
Matthias Raschendorfer
CLM-Training Course 2011