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##### Boundary Layer Parameterization

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**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 non-local closure for such a parameterization?**Typical Boundary Layer Structure**surface layer laminar sublayer sunrise 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 mechanicallyturbulent. • 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 vegetatedsandy 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!**Daytime vs. Nighttime Turbulence**(analogous to vertical motion) • Nocturnal turbulence, primarily associated with small-scale turbulent eddies, is of higher frequency but lower amplitude. • Daytime turbulence, associated with both small- and large-scale 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 laminar sublayer sunrise 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.**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) = 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… 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… • Expand and make the assumption that ρ’ << ρ-bar…**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… • The Reynolds’ stress term represents the effects of small-scale turbulence on the large-scale motion. Reynolds’ stress**Boundary Layer Parameterization**• The quantity 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 • 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… • 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: • 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., • This substitution and subsequent manipulation gives the complete predictive equation for the covariance.**Boundary Layer Parameterization**• Predictive equation for the covariance term(s)… • Six predictive equations for • 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., • 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 resolved-scale 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.),**Local Closure**• The covariance, or flux, of ξ is parameterized by… • For positive K, the flux is down the local gradient of (i.e., higher values transported toward lower values). • Combining the two equations, is the resolved-scale, local value of ξ K is a scalar with units of m2 s-1**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. value at the reference grid point value at some donor grid point, where b = 1->N, with N being the total number of PBL grid points