Boundary Layer Parameterization
14 November 2012
Thematic Outline of Basic Concepts
• What are the relevant physical processes that must
be parameterized?
• How are these processes parameterized within a
boundary layer parameterization?
• What is the difference between a local and a nonlocal closure for such a parameterization?
Typical Boundary Layer Structure
surface layer
sunrise
laminar
sublayer sunset
Overview
• Boundary Layer: Layer over which the influence of
the surface is directly transmitted to the free
atmosphere.
• The boundary layer is a turbulent, mixed layer,
characterized by small-scale turbulent eddies.
• Turbulent eddies transport (or mix) heat, moisture,
and momentum within the boundary layer.
Overview
• Heat and moisture are mixed upward from the
surface, where they are extracted via surface fluxes.
• Momentum is typically mixed downward in an
environment with wind speed increasing with height.
• Turbulent eddies also transmit the frictional stress
exerted by the surface upon the atmosphere.
Turbulence
• There are two causes of turbulence…
– Buoyancy
– Vertical shear of the horizontal wind
• Buoyancy refers to the local instability created by
daytime heating of the underlying land surface.
• Dry convective plumes, both upward and downward,
result from the release of this instability.
Turbulence
• The vertical extent of the daytime mixing defines the
depth of the daytime boundary layer.
• At night, heating of the surface ends and the
underlying surface cools. This creates a shallow
stable, or inversion, layer near the ground.
• Any turbulent energy within the nocturnal boundary
layer must be extracted from the vertical wind shear.
– Typically minimal unless the horizontal wind is strong.
Turbulence
• When vertical wind shear is small and there is no
buoyancy, the flow is laminar, or nonturbulent.
• For larger vertical wind shear, the flow becomes
mechanically turbulent.
– Predominant during the nighttime hours.
• In the presence of buoyancy, the flow becomes
convectively turbulent.
– Dominant during the daytime hours.
Turbulence
• Since the wind speed perpendicular to any rigid
surface like the ground must be zero, turbulence
cannot exist at the surface.
• Without turbulence, some other means of
transporting water vapor and heat energy from the
surface into the boundary layer must exist.
• This occurs in a shallow (thickness of millimeter or
less) layer known as the laminar sublayer.
Turbulence
• In the laminar sublayer, heat, moisture, and frictional
effects are communicated on a molecular level.
• Molecular mixing reduces the magnitude of the
vertical gradients of heat, moisture, and momentum.
– Transport is down-gradient (from higher to lower values).
• Molecular mixing is far less efficient than turbulent
mixing, however.
Turbulence
• Within the lowest 50 – 100 m of the boundary layer,
turbulent transports vary little with height compared
to their variability in the mixed layer above.
• The layer over which this is the case is known as the
surface layer.
Overview of Boundary Layer Structure
• Laminar sublayer: interface between the ground and
the surface layer.
• Surface layer: interface between the laminar
sublayer and boundary layer.
• Boundary layer: interface between the surface layer
and the free atmosphere.
Boundary Layer Parameterization
• A boundary layer parameterization must be able to
accurately represent boundary layer processes.
• A surface layer parameterization must be able to
communicate information between the underlying
surface and mixed layer in a model simulation.
• A land-surface parameterization encapsulates the
impact of the surface, from the laminar sublayer to
soil levels below ground, upon the atmosphere.
Why Boundary Layer Parameterization?
• Turbulent eddies occur on spatiotemporal scales of
centimeters to meters and seconds to minutes.
– True for both horizontal and vertical length scales.
• Molecular mixing and transport processes occur on
even smaller scales.
• Local variability in boundary layer processes remains
somewhat unknown.
– e.g., roughness length and other surface contrasts
Boundary Layer as a Mixed Layer
salt flat
time
vegetated
sandy site
Vertical profile of potential temperature becomes progressively more uniform with time!
Boundary Layer as a Mixed Layer
Vertical profiles of u, θ, and ρv are largely uniform (constant) with height in the mixed layer!
(analogous to
vertical motion)
Daytime vs. Nighttime Turbulence
• Nocturnal turbulence, primarily associated with small-scale
turbulent eddies, is of higher frequency but lower amplitude.
• Daytime turbulence, associated with both small- and largescale turbulent eddies, reflects the superposition of the high
and low frequency modes with overall greater amplitude.
Nocturnal Boundary Layer
• Mixing in the nocturnal boundary layer can be
intermittent or oscillatory in nature.
– Vertical wind shear exceeds some critical value.
– Mixing reduces the vertical shear to a subcritical value.
• Above the nocturnal stable layer, residual turbulence
remains from the daytime boundary layer.
– Typically decays in depth and in intensity with time.
– Cause of this decay: internal friction (not surface friction).
Typical Boundary Layer Structure
surface layer
sunrise
laminar
sublayer sunset
• The depth of mixed layer can be > or < the 1 km shown above
and is proportional to the heating of the underlying surface.
• Influences upon surface heating include meteorological
conditions, seasonality, continentality, and land-surface
structure.
Surface and Inversion Layers
Inversion Layer Structure
• Potential temperature increases with height, creating
a stable thermodynamic situation.
• Horizontal wind speed increases with height.
– Impact of surface friction becomes negligible
– Dashed line, Fig. 4.12b: theoretical wind speed w/o friction
• Moisture decreases with height.
– Entrainment within the inversion mixes in drier air from
above the boundary layer.
Surface Layer Structure
• Potential temperature decreases with height due to
strong sensible heating of the underlying surface.
– Superadiabatic lapse rate; unstable.
• Horizontal wind speed increases with height.
– Wind profile is a log-wind profile (more shortly).
• Moisture decreases with height.
– Surface latent heat flux keeps surface moisture content
relatively high.
Surface Layer Structure
• Boundary layer structure is strongly impacted by the
characteristics of the underlying land-surface.
• This is particularly manifest via the roughness of that
surface, or the roughness length (z0).
– Smooth surfaces have relatively small roughness lengths.
– Rougher surfaces have larger roughness lengths.
– Typical length scales: O(mm) to O(cm) or larger.
Surface Layer Structure
• The roughness length impacts the vertical structure
of the horizontal wind within the surface layer.
• This subsequently impacts the near-surface
turbulent fluxes of heat, moisture, and momentum.
• For a well-mixed boundary layer, it can be shown:
u*  z 
u  z   ln 
k  z0 
u(z) = speed of the mean horizontal wind at a height z above the ground
u* = friction velocity (not a function of height)
k = von Karman constant (typically 0.35 – 0.4)
Surface Layer Structure
• The friction velocity represents the drag of the
atmosphere against the surface (the frictional stress).
• The roughness length is the height at which the
mean wind speed goes to zero in neutral conditions.
• As u* and k are both constant with height, the mean
horizontal wind speed increases logarithmically with
increasing height.
Surface Layer Structure
Note that if the assumption of neutrally stable does not hold,
qualitatively similar results are obtained (dashed lines).
Boundary Layer Non-Uniformity
• Boundary layer structure is not always as uniform as
the foregoing discussion may make it seem.
• Potential causes of non-uniformity...
– Non-uniform spatial and size distributions of aerosols can
impact radiative transfer within the boundary layer.
– Horizontal contrasts in land-surface properties, drastic or
subtle, can lead to internal boundary development,
Boundary Layer Non-Uniformity
• Examples of internal boundary development...
– Boundary layer air moving from a drier, warmer, smoother
surface to a moister, cooler, rougher one.
– Downstream advection of a mixed layer that forms due to
heating over elevated terrain.
– Downstream advection over a cooler body of water of a
mixed layer that formed over hot, dry land surfaces.
• The drier, warmer boundary layer will lift over the
moister, cooler one, forming an internal boundary.
Boundary Layer Non-Uniformity
Schematic of an elevated mixed layer over the Great Plains.
Boundary Layer Parameterization
• How do we parameterize all of these processes and
structures within a numerical model?
• We start by examining the primitive equations to
demonstrate the terms that must be parameterized.
– We’ll do this explicitly only for the momentum equations,
but note that the same is also done for other equations.
• Subsequently, we discuss different mathematical
constructs for the parameterization of these terms.
Boundary Layer Parameterization
• Momentum equations, in tensor notation…
ui
u
1 p
 u j i   i 3 g  f ij3u j 
t
x j
 xi
i, j = (1, 2, 3), with j independent of i
ui  u1 = u, u2 = v, u3 = w
xi  x1 = x, x2 = y, x3 = z
δij is 1 when i = j but zero otherwise
εijk = 1 if i, j, k are in ascending order
-1 if i, j, k are in descending order
0 if any of i, j, k are equal to one another
Boundary Layer Parameterization
• Separate dependent variables into mean (resolved)
and perturbation (turbulent) components…

 
 


 




1

ui  ui'   u j  u 'j
ui  ui'   i 3 g  f ij3 u j  u 'j 
p  p'
t
x j
   ' xi

• Expand and make the assumption that ρ’ << ρ-bar…
'
ui ui'
ui
ui'

u

u
1  p 1 p'

 u j
uj
 u 'j i  u 'j i   i 3 g  f ij3 u j  f ij3u 'j 

t
t
x j
x j
x j
x j
 xi  xi
Boundary Layer Parameterization
• Take the Reynolds average of this equation and
simply the result using Reynolds’ postulates and the
continuity equation (Eqn. 2.14, text pg. 9) to obtain…

 ui
u
1  p  ui ' u j '
 u j i   i 3 g  f ij 3 u j 

t
x j
x j
 xi

Reynolds’ stress
• The Reynolds’ stress term represents the effects of
small-scale turbulence on the large-scale motion.
Boundary Layer Parameterization
• The quantity ui ' u j ' is called a double correlation, or a
second statistical moment. It is also often known as
a covariance.
• The momentum equations only predict the mean
values of u, v, and w. We need to somehow define ui ' u j '
– Parameterize in terms of the resolved-scale (mean) fields
– Develop predictive equations for the second-moment term
Boundary Layer Parameterization
• To develop a predictive equation, subtract our most
recent equation from the one that preceded it…

'
ui'
ui'
1 p '  ui ' u j '
'  ui
' ui
'
 u j
uj
uj
 f ij3u j 

t
x j
x j
x j
x j
 xi

• This gives a predictive equation for the turbulent
velocity components.
Boundary Layer Parameterization
• A predictive equation for the covariance between ui
and uk (or, likewise, ui and uj) is given by:





ui ' uk '  ui ' uk '  uk ' ui '
t
t
t
• This is a form of the partial derivative product rule.
– Previous slide: an equation for part of the second right-hand side term
– We can substitute k for i in that equation to get an expression for part
of the first right-hand side term
Boundary Layer Parameterization
• After doing so, we can manipulate the equation on
the previous slide…
– Multiplication by uk’ or ui’, Reynolds averaging, Reynolds’
postulate application, substitute from continuity
– Note: substitution from continuity involves making use of
the product rule, i.e., a b  b a  ab






• This substitution and subsequent manipulation gives
the complete predictive equation for the covariance.
Boundary Layer Parameterization
• Predictive equation for the covariance term(s)…




 
 






ui ' uk '  u j
ui ' u k '  ui ' u j '
uk  uk ' u j '
ui 
ui ' u k ' u j '
t
x j
x j
x j
x j

 ui ' uk ' 
1 


 f  kj3 ui ' u j '   ij3 uk ' u j '  
p' uk ' 
p ' ui '  p' 


x

x

x

x
  i
k
i 
 k







• Six predictive equations for u' u', u' v', u' w', v' v', v' w', w' w'
• One problem, though – there’s a triple correlation
term that shows up in the above.
Boundary Layer Parameterization
• We could then create predictive equations for that
triple correlation term, containing ten unknowns, but
that would contain a quadruple correlation term!
• This concept is what ultimately defines the
turbulence parameterization problem.
– First order closure: predictive equations for the state variables,
parameterization for the covariances.
– Second order closure: predictive equations for the state variables and
the covariances, parameterization for the triple correlations.
– Third order closure: predictive for the state variables, covariances, and
triple correlations, parameterization for the quadruple correlations.
Boundary Layer Parameterization
• The order of the closure is determined by the
highest-order correlations for which predictive
equations exist.
• See also: Tables 4.1 and 4.2 in the text (pg. 150).
• Higher-order closures are thought to be more
accurate because they explicitly predict more
statistical moments.
Boundary Layer Parameterization
• The higher of moments that are predicted, the less
that the parameterization of the next-higher
moment will impact the state (first moment) fields.
• Other non-integer closures are possible.
– Ex: parameterize some second-order moments and predict
other second-order moments.
– This defines a 1.5-order closure method.
– Other non-integer closures are possible (2.5, etc.)
Boundary Layer Parameterization
• How do we actually parameterize those higher order
moments, however?
• There are two different approaches to do so…
– Local Closure: define the unknown quantity from known
quantities or their vertical derivatives at the same grid point.
– Non-Local Closure: estimate the unknown quantity in terms
of known quantities elsewhere in the vertical.
– Non-local and higher-order closures are generally more
accurate but more computationally expensive and complex.
– Almost all current parameterizations are non-local closures.
Boundary Layer Parameterization
• Second-order or higher closure methods allow for
the quantification of the total turbulent kinetic
energy (TKE) from the covariances, i.e.,
TKE 1
 ui ui
mass 2
• There exist many different local and non-local closure
approximations; we only discuss select examples.
Local vs. Non-Local Closure
local
two non-local examples
depends on resolved-scale field
only at a grid point or vertical
derivatives thereof (thus also being
influenced by adjacent points)
can be influenced by resolved-scale field at a grid
point but can also be influenced by a resolvedscale field at grid points well-removed from the
grid point under consideration
Local Closure
• Works best when eddies are small (locally-contained)
• Example: K-theory, or gradient-transport theory
• In a first-order closure, the second moments are
parameterized.
• For some generic variable ξ (i.e., u, v, w, T, qv, etc.),



 ... 
 'u j '
t
x j

Local Closure
• The covariance, or flux, of ξ is parameterized by…
 ' u j '  K



x j
 is the resolved-scale, local value of ξ
K is a scalar with units of m2 s-1
• For positive K, the flux is down the local gradient of 
(i.e., higher values transported toward lower values).
• Combining the two equations,


 ... 
t
x j



K
 
 x
j



Local Closure
• This closes the system of equations as all variables
have either a prognostic or diagnostic equation for
their values; in other words, there are no unknowns.
• The coefficient K is known as the eddy diffusivity,
eddy viscosity, or eddy transport coefficient.
• Often, separate values of K are used for heat,
moisture, and momentum in the parameterization.
Local Closure
• The values of K control the turbulent flux.
• Intuition suggests that K should be linked to factors
that produce turbulence, namely buoyancy and
vertical wind shear.
• However, many different methods for determining K
exist, whether based on the above or not.
Non-Local Closure
• Much of the mixing the boundary layer can be
associated with large (O(km)) eddies.
• These eddies are not related to the local static
stability or vertical wind shear at some grid point.
• Instead, these eddies are driven by the mean stability
of the entire boundary layer, which in turn responds
to the surface heat flux.
Non-Local Closure
• Approach 1: transilient turbulence theory (Fig 4.15b)
• Each grid point in the PBL is influenced by turbulent
eddies of varying sizes that mix information from
other grid points into that grid point.
• Again, let  refer to some variable.
a
value at the reference grid point
b
value at some donor grid point, where b = 1->N,
with N being the total number of PBL grid points
Non-Local Closure
• We define a variable cab, or the fraction of air at point a
that has been transported from point b over Δt.
• The exchange between a and each b is given by…
N
 a t  t    cab tb t 
b 1
• cab has dimensions N x N, since all b can also be an a,
and is a function of the turbulence.
– Thus, it is the same for all possible 
– Parameterization problem requires appropriately defining cab
Non-Local Closure
• Approach 2: Blackadar method (Fig 4.15c)
• Vertical exchange takes place between each PBL grid
point and the lowest grid point.
– Presumes that both small and large eddies alike have their
roots within the surface layer.
• Intensity of the daytime vertical exchange is
determined from the surface heat flux and the
thermal structure of the entire mixed layer.
Non-Local Closure
• Example turbulent transfer term:

 ...  m a   
t
θ = potential temperature at some PBL grid point
θa = potential temperature at the top of the surface layer
m = function of the surface heat flux
• m represents the fraction of mass in a grid box in the
column exchanged with the surface grid box over Δt.
– Parameterization problem requires appropriately defining m.
• This is similar to the transilient turbulence problem,
but with less dimensionality.
Local vs. Non-Local Closure
Local static stability determined
from local change in θv with height.
Non-local static stability determined from
change in θv with height over the entire PBL.
Turbulent vs. laminar flow is
inherently dependent on vertical
wind shear and buoyancy.
Local vs. Non-Local Closure
• The local vertical heat flux can be defined from:

 'u j '  K


x j
 v ' w'   K

v
z
 
• This is thus defined by the vertical gradient of θv.
• The non-local interpretation of the vertical heat flux
is different and inherently more complex.
Local vs. Non-Local Closure
The local interpretation of the
The non-local interpretation of the vertical heat The observed heat flux is mostly
vertical heat flux is dependent
flux enables heat to flow from hot to cold, with
upward, since θv generally
upon the local vertical change of θv, its magnitude and direction dependent upon the decreases with height, except at
just like the local static stability.
specific non-local mixing details.
the inversion layer.