9.4 Monin-Obukhov similarity theory

Monin-Obukhov similarity theory is a cornerstone of atmospheric boundary layer physics. It provides a framework for understanding turbulent flows near the Earth's surface, using dimensional analysis to describe vertical turbulence structure.

The theory assumes a homogeneous surface layer with constant fluxes and negligible Coriolis effects. It uses dimensionless groups to construct universal functions, leading to the logarithmic wind profile and stability corrections for non-neutral conditions.

Fundamentals of Monin-Obukhov theory

• Monin-Obukhov similarity theory provides a framework for understanding turbulent flows in the atmospheric surface layer
• Applies dimensional analysis to describe the vertical structure of turbulence near the Earth's surface
• Forms the basis for many boundary layer parameterizations used in atmospheric models

Key assumptions

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• Surface layer is horizontally homogeneous with constant fluxes
• Turbulent fluxes dominate over molecular diffusion
• Coriolis force effects are negligible in the surface layer
• Flow is statistically stationary over the averaging period
• Assumes a flat, uniform surface with no obstacles

Dimensional analysis approach

• Uses Buckingham Pi theorem to derive dimensionless groups
• Identifies relevant physical parameters (friction velocity, buoyancy flux, height)
• Constructs universal functions based on dimensionless ratios
• Leads to logarithmic wind profile in neutral conditions
• Incorporates stability corrections for non-neutral atmospheres

Obukhov length scale

• Fundamental length scale in Monin-Obukhov theory
• Defined as $L = -\frac{u_*^3}{\kappa(g/\theta_v)H_0/\rho c_p}$
• Represents the height where buoyancy production equals shear production
• Negative values indicate unstable conditions, positive values stable conditions
• Magnitude indicates the strength of stability effects
  • Large |L| suggests near-neutral conditions
  • Small |L| indicates strong stability influence

Similarity functions

• Describe how atmospheric variables deviate from neutral conditions
• Express turbulent fluxes and gradients as functions of stability parameter z/L
• Enable prediction of vertical profiles in the surface layer

Momentum similarity function

• Denoted as ϕm(z/L), describes wind shear in non-neutral conditions
• Approaches 1 in neutral conditions (z/L → 0)
• Decreases in unstable conditions (z/L < 0) due to enhanced mixing
• Increases in stable conditions (z/L > 0) due to suppressed turbulence
• Often expressed as ϕm(z/L) = (1 - αz/L)^-β for unstable conditions

Heat similarity function

• Represented by ϕh(z/L), describes temperature gradient in non-neutral conditions
• Approaches Prandtl number (≈ 0.74) in neutral conditions
• Decreases more rapidly than ϕm in unstable conditions
• Increases more steeply than ϕm in stable conditions
• Can be expressed as ϕh(z/L) = β(1 - γz/L)^-1/2 for unstable conditions

Moisture similarity function

• Denoted as ϕq(z/L), describes water vapor gradient in non-neutral conditions
• Generally assumed to be equal to ϕh(z/L) in most applications
• Reflects similarity between heat and moisture transport in turbulent flows
• May deviate from ϕh in very stable or unstable conditions
• Crucial for estimating evaporation and latent heat fluxes

Stability parameters

• Quantify the relative importance of buoyancy and shear in turbulence production
• Used to classify atmospheric stability and determine appropriate similarity functions
• Essential for parameterizing turbulent fluxes in numerical models

Richardson number

• Dimensionless ratio of buoyancy to shear production of turbulence
• Gradient Richardson number defined as $Ri = \frac{g}{\theta}\frac{\partial\theta/\partial z}{(\partial U/\partial z)^2}$
• Negative values indicate unstable conditions, positive values stable conditions
• Critical value (Ri ≈ 0.25) often used as threshold for turbulence suppression
• Difficult to measure directly due to required vertical gradient measurements

Bulk Richardson number

• Simplified version of Richardson number using finite differences
• Calculated as $Ri_B = \frac{g\Delta z\Delta\theta}{\theta(\Delta U)^2}$
• Easier to compute from standard meteorological measurements
• Used in many numerical weather prediction models
• May not accurately represent local stability in strongly stratified layers

Flux Richardson number

• Ratio of buoyancy flux to shear production of turbulent kinetic energy
• Defined as $Ri_f = \frac{(g/\theta)\overline{w'\theta'}}{\overline{u'w'}\partial U/\partial z}$
• Directly related to turbulent fluxes rather than mean gradients
• More physically relevant for describing turbulence dynamics
• Challenging to measure due to required eddy covariance measurements

Flux-profile relationships

• Connect turbulent fluxes to mean vertical gradients in the surface layer
• Form the basis for estimating surface fluxes from routine meteorological measurements
• Incorporate stability corrections through universal functions

Momentum flux profile

• Relates momentum flux to wind speed gradient
• Expressed as $\frac{\kappa z}{u_*}\frac{\partial U}{\partial z} = \phi_m(z/L)$
• Integrates to logarithmic wind profile with stability correction
• Used to estimate surface stress and friction velocity
• Crucial for modeling wind profiles in the atmospheric boundary layer

Heat flux profile

• Connects sensible heat flux to potential temperature gradient
• Given by $\frac{\kappa z}{\theta_*}\frac{\partial \theta}{\partial z} = \phi_h(z/L)$
• Integrates to logarithmic temperature profile with stability correction
• Allows estimation of surface sensible heat flux
• Important for understanding thermal structure of the boundary layer

Moisture flux profile

• Links water vapor flux to specific humidity gradient
• Formulated as $\frac{\kappa z}{q_*}\frac{\partial q}{\partial z} = \phi_q(z/L)$
• Assumes similarity between heat and moisture transport
• Enables estimation of surface latent heat flux and evaporation
• Critical for modeling water vapor distribution and cloud formation

Surface layer scaling

• Provides characteristic scales for velocity, temperature, and moisture in the surface layer
• Allows normalization of turbulent quantities for universal representation
• Facilitates comparison of measurements from different sites and conditions

Velocity scales

• Friction velocity (u*) serves as primary velocity scale
• Defined as $u_* = \sqrt{-\overline{u'w'}}$
• Represents intensity of turbulent momentum transport
• Used to normalize wind speed profiles and turbulence statistics
• Typically ranges from 0.1 to 1 m/s in the atmospheric surface layer

Temperature scales

• Temperature scale (θ*) characterizes turbulent heat transport
• Defined as $\theta_* = -\overline{w'\theta'}/u_*$
• Used to normalize temperature profiles and heat flux measurements
• Negative in unstable conditions, positive in stable conditions
• Magnitude typically ranges from 0.01 to 1 K in the surface layer

Moisture scales

• Specific humidity scale (q*) represents turbulent moisture transport
• Defined analogously to temperature scale: $q_* = -\overline{w'q'}/u_*$
• Used to normalize humidity profiles and latent heat flux measurements
• Positive for upward moisture flux (evaporation), negative for downward flux
• Magnitude depends on surface moisture availability and atmospheric conditions

Limitations and extensions

• Monin-Obukhov theory has known limitations in certain atmospheric conditions
• Various extensions and modifications have been proposed to address these issues
• Understanding these limitations critical for proper application of the theory

Validity in different conditions

• Theory works best in near-neutral and moderately unstable conditions
• Breaks down in strongly stable conditions (z/L > 1) due to intermittent turbulence
• May not apply in very unstable conditions (free convection limit)
• Assumes horizontal homogeneity, limiting applicability over complex terrain
• Requires steady-state conditions, challenging in rapidly changing weather

Non-dimensional gradients

• Universal functions may vary between sites and stability ranges
• Different formulations proposed for stable and unstable conditions
• Some researchers suggest separate functions for momentum, heat, and moisture
• Ongoing debate about the exact form of stability functions in very stable conditions
• Recent studies explore non-local effects on gradient-flux relationships

Roughness sublayer effects

• Theory assumes measurements above the roughness sublayer
• Roughness sublayer depth varies with surface characteristics (typically 2-5 times canopy height)
• Additional corrections needed for flux-profile relationships within roughness sublayer
• Affects flux footprint calculations and interpretation of near-surface measurements
• Important consideration for flux measurements over forests and urban areas

Applications in atmospheric modeling

• Monin-Obukhov theory forms the basis for many surface layer parameterizations
• Widely used in numerical weather prediction and climate models
• Enables estimation of surface fluxes from routine meteorological observations

Boundary layer parameterization

• Provides lower boundary conditions for planetary boundary layer schemes
• Used to calculate surface drag, heat flux, and moisture flux
• Incorporates stability-dependent eddy diffusivity profiles
• Influences vertical mixing and turbulent transport throughout the boundary layer
• Critical for accurate representation of near-surface weather conditions

Surface flux estimation

• Allows calculation of momentum, heat, and moisture fluxes from standard measurements
• Utilizes bulk aerodynamic formulas based on Monin-Obukhov similarity
• Requires input of surface roughness length and stability functions
• Widely used in agricultural meteorology and hydrology applications
• Forms basis for evapotranspiration estimation in land surface models

Turbulence closure schemes

• Provides scaling relationships for higher-order turbulence closure models
• Used to parameterize turbulent kinetic energy and dissipation rate profiles
• Informs eddy viscosity and diffusivity formulations in k-ε and k-ω models
• Helps constrain turbulence length scales in mixing length approaches
• Crucial for representing subgrid-scale processes in large-scale atmospheric models

Experimental validation

• Extensive efforts to validate Monin-Obukhov theory through various experimental approaches
• Combination of field measurements, laboratory studies, and numerical simulations
• Ongoing research to refine and extend the theory based on observational evidence

Field measurements

• Eddy covariance techniques used to directly measure turbulent fluxes
• Flux-gradient methods employed to test similarity functions
• Tall tower measurements provide vertical profiles in the surface layer
• Aircraft observations used to study spatial variability and heterogeneity effects
• Long-term datasets (FLUXNET, AmeriFlux) enable validation across diverse ecosystems

Wind tunnel studies

• Controlled experiments to isolate specific processes and parameters
• Allow systematic variation of stability conditions and surface characteristics
• Used to study roughness sublayer effects and complex terrain influences
• Provide detailed measurements of turbulence statistics and spectra
• Help validate similarity functions and flux-profile relationships

Large eddy simulations

• Numerical experiments to study surface layer dynamics at high resolution
• Enable investigation of processes difficult to measure in the field
• Used to test assumptions of horizontal homogeneity and constant flux layer
• Provide insights into non-local effects and internal boundary layer development
• Help refine parameterizations for coarser-resolution atmospheric models

Monin-Obukhov vs other theories

• Monin-Obukhov theory complements and extends other approaches to boundary layer turbulence
• Important to understand relationships and differences between various theoretical frameworks
• Each approach has strengths and limitations for different applications

Comparison with K-theory

• K-theory assumes downgradient diffusion with constant or height-dependent eddy diffusivity
• Monin-Obukhov theory provides stability-dependent scaling for eddy diffusivity
• K-theory simpler to implement but lacks universal applicability across stability ranges
• Monin-Obukhov approach captures non-local effects through similarity functions
• Hybrid approaches combine K-theory with Monin-Obukhov scaling in some models

Relation to mixing length theory

• Mixing length theory assumes turbulent eddies have characteristic length scale
• Monin-Obukhov theory incorporates stability effects on effective mixing length
• Obukhov length serves as stability-dependent limit on mixing length in stable conditions
• Mixing length approaches often use Monin-Obukhov scaling in the surface layer
• Both theories contribute to development of more advanced turbulence closure schemes

Recent developments

• Ongoing research continues to refine and extend Monin-Obukhov similarity theory
• New approaches address limitations and expand applicability to complex conditions
• Incorporation of advanced measurement techniques and high-resolution modeling

Non-local effects

• Recognition of importance of large-scale eddies in unstable conditions
• Development of convective velocity scale and mixed-layer similarity
• Inclusion of entrainment effects at the top of the boundary layer
• Exploration of non-local flux-gradient relationships in strongly unstable conditions
• Incorporation of top-down and bottom-up diffusion concepts

Heterogeneous surfaces

• Extension of theory to account for surface heterogeneity and patchiness
• Development of blending height concept for transitions between surface types
• Study of internal boundary layer development over changing surface conditions
• Incorporation of footprint models to interpret flux measurements over heterogeneous terrain
• Exploration of mosaic approaches for subgrid-scale surface variability in models

Stable boundary layer modifications

• Recognition of limitations of traditional theory in very stable conditions
• Development of z-less scaling for strongly stable stratification
• Incorporation of intermittency and non-stationarity in flux-profile relationships
• Exploration of anisotropic turbulence effects in stable boundary layers
• Investigation of low-level jets and their impact on surface layer structure