26 Atmospheric Turbulence

M K Nanda

1. Learning outcomes
2. Introduction
3. Turbulence spectra
4. Eddy transport
5. Conservation equations in the PBL
5.1. The closure problem
6. Prandtl’s mixing length theory
7. Turbulence kinetic energy
8. Stability and turbulent nature of BL
9. The Ekman layer
10. Summary

1. Learning outcomes

• After studying this module, you shall be able to:
• Have general understanding on atmospheric turbulence
• Learn about turbulence spectra in the atmospheric boundary layer
• Understand turbulent transport of energy and mass in the boundary layer
• Become familiar with conservation equations in the boundary layer and closure problem Understand turbulent kinetic energy and its budget in the boundary layer
• Understand Ekman layer concept

2. Introduction

A turbulent flow is characterized by irregular/ random features of the variables (for e.g. velocity, temperature, water vapor or tracer concentration), over a wide range of scales of motion (in terms of space and time), and by mixing properties.

Reynolds number (Re), a dimensionless number given by UL/ν, is commonly used to characterize the turbulent/ laminar nature of the flow of a viscous fluid. Here U is mean velocity, L the length scale, for e.g. the depth of the fluid, and ν is the kinematic viscosity of the fluid representing internal molecular friction. If Re < 2000, the flow is expected to be laminar (and predictable), whereas flow will be turbulent for Re > 4000. Re in the range 2000 – 4000 corresponds to the transition. For large values of Re, the flow accelerations can be quite large, and only the statistical description of the turbulent flow will make sense for many practical purposes. Re in the atmospheric boundary layer is always very large implying existence of turbulent conditions. A time series of u and w components of wind velocity, air temperature and CO2 concentration in the surface layer is given in Fig. 1 to illustrate the random/ chaotic features of the flow.

An important feature of the turbulent flow is the presence of a large range of time and space scales of the motion in the form of organized eddies. The high frequency fluctuations are usually superimposed on organized longer time scales. The temporal scales of eddies range from fraction of a second to several minutes and the associated spatial scales are from few millimeters to several hundred meters. The largest dimensions are generally decided by the boundaries of the medium and the driving forces. Space scales smaller than 3 km and time scales less than 1 hour are on the micro scale and considered to be part of atmospheric turbulent motion. Turbulence in the atmospheric boundary layer is mainly generated by solar heating of the surface and wind shear. Of the turbulent eddies, larger ones are more energetic. Turbulent flows transport momentum, energy and mass without a mean velocity component. This is particularly important for the vertical exchange as the mean vertical velocity is close to zero in the lower part of the atmosphere.

Turbulence is critically important for the energy cascade that occurs in the atmosphere.

• Energy associated with large-scale mean motion is transferred to the larger turbulence scales (i.e., the large eddies).
• These dynamically unstable large eddies (of the scale up to the depth of the PBL) break down into progressively smaller eddies via an inertial cascade.
• It is called a cascade because eddies are deformed and folded most efficiently by other eddies of comparable scales, and this squeezing and stretching transfers energy between nearby length scales. Thus the large eddies feed energy into smaller ones, and so on until the eddies become so small as to be viscously dissipated.
• This cascading breakdown of eddy size continues until eddies are so small that energy is consumed by work against viscous forces that convert kinetic energy into heat.

Turbulence also exist above the boundary layer in convective clouds as well as in thin layers of large horizontal extent associated with jet streams, known as Clear Air Turbulence (CAT).

3. Turbulence spectra

The turbulence power spectrum can be generated from a time series of discrete measurements of atmospheric variables (e.g. wind components, air temperature, etc.) with the help of Fast Fourier Transform (FFT) or wavelet transform. The representation of spectral density (square of the amplitude of FFT or wavelet transform amplitude) versus frequency f (or wave number k = 2πf/u, u the mean horizontal wind speed) gives the turbulence power spectrum. The resulting power spectrum can be interpreted as the variance of the variable due to eddies with wave number between k and k+dk.

The turbulent energy spectrum, shown below has three distinct regions:

(A) the region of energy production

(B) the inertial subrange

(C) the dissipation region

1. The energy-producing range:

• Turbulence is produced by buoyancy or shear, and the fluctuations have a characteristic size referred as integral length scale Λ.
• For a convective boundary layer, Λ would be approximately the depth of the boundary layer, while for shear production it would be close to the wavelength of the developing Kelvin-Helmholtz waves.

1. The inertial subrange:

Within the spectra of turbulence, there exists a spectral region independent of both the turbulence production and viscous dissipation. In this region, the intensity of turbulence at a given wave number is only dependent on the wave number and the turbulent dissipation rate.

•  Energy is neither produced nor dissipated in this range of turbulent eddy sizes, but it is handed down to smaller and smaller scales through turbulent energy cascade.

•  The turbulence in the inertial subrange is 3-dimensional, isotropic (i.e. the structures have no preferred orientation), homogeneous and self-similar.

•  Local isotropy also implies vanishing of all correlations between velocity components and between velocity components and scalars. No turbulent fluxes in the inertial subrange.

•  In the inertial subrange, the turbulent energy spectrum E(κ) depends only on the wave number κ (in m-1) and the turbulent kinetic energy (TKE) dissipation rate ε (in m2 s-3) – known as the Kolmogorov similarity theory. where α is the Kolmogorov constant.

    C.The dissipation range

The kinetic energy of parcels of air is converted into the kinetic energy of air molecules, i.e. TKE is dissipated into heat. The length scale below which this occurs depends only on the TKE dissipation rate ε ( in m2 s-3 ) and the kinematic viscosity ( ν ) in m2s-1. In the dissipation range, the scaling lengthKolmogorov microscale is typically around 1 mm. It is the scale at which eddies dissipate into heat. Kolmogorov microscale determines the size of the smallest eddies that can be sustained in presence of viscosity.

4. Eddy transport

The transport/ exchange is usually expressed in terms of the flux of the quantity of interest, which is defined as the quantity per unit area per second. e.g., heat flux, water vapor flux, momentum flux, CO2 flux. The product of quantity and flow  will have three components associated with three components of flow (u, v, w). Momentum having three components by itself, the momentum flux will have nine components.

Turbulent eddies are responsible for most of the vertical transport of energy and mass between the earth’s surface and the boundary layer, as the mean vertical velocity is nearly zero close to the surface. The eddy flux concept is explained with the help of schematic diagram (Fig. 3) for kinematic sensible heat flux during daytime. As shown in the figure, potential temperature in the surface layer typically decreases with height. The turbulent eddy motions in the surface layer (and throughout the boundary layer), responsible for the transport, consist of both up and down motions. At one moment, eddy moves air parcel upward and the next moment another eddy may move air parcel downward. Each air parcel will have its own characteristics, such as temperature, humidity, gas concentration, etc. and will be moving with a characteristic speed defined by eddy motion. In the example shown, the air parcel going up from lower level will introduce a +ve fluctuation in the temperature whereas an parcel moving down from above will introduce –ve fluctuation as it crosses the sensor at a certain height above surface. Together with the positive and negative sign of vertical velocity fluctuation, the product with temperature fluctuation will be positive heat flux, both for up and down motions for majority of the eddy motions during daytime. The nighttime will have negative heat fluxes as air temperature increases with height in the stable boundary layer.

5. Conservation equations in the PBL

The original conservation equations for momentum, mass and energy are developed for the whole of earth’s atmosphere considering only mean flow conditions. In the atmospheric boundary layer,  turbulent transport is very significant, thus the conservation equations governing atmospheric motions must be modified to represent the turbulent exchange.

The steps involved in deriving equations in Boundary Layer are:

• Identify the governing equations for boundary layer
• Apply Boussinesq Approximation
• Expand the dependent variables into mean and turbulent parts using Reynolds decomposition
• Apply Reynolds averaging to get equations for mean variables within turbulent flow

In Boussinesq approximation, density is assumed to be a constant mean value (ρ ≈ ρ0 ), except when it contributes directly to the buoyancy, i.e. except in the buoyancy term in the vertical momentum equation. The approximation is based on the fact that air density typically does not change more than 10% in the atmospheric BL.

Following the steps mentioned above, the modified equations for mean state variables in the BL are given by:

The terms in square brackets in the above equations represent gradients of turbulent fluxes. The equations indicate that turbulence contributions to changes in different variables are from flux divergence.

• In the boundary layer, the magnitude of the turbulent flux divergence terms are often of the same or higher order of magnitude as the pressure gradient, Coriolis and friction terms, and it is not possible to neglect them even when only the mean flow is of direct interest.
• Outside the boundary layer, the turbulent fluxes are mostly weak so that the terms in square brackets can be neglected in the analysis of large-scale flow.

5.1. The closure problem

The conservation equations in the BL discussed above have more number of unknowns than the number of equations, hence they cannot be solved as it is. The additional unknown terms are the flux gradient terms. If we introduce new equations (prognostic or diagnostic) to solve for these unknowns (e.g. for u′w′), additional unknowns will be added (e.g. terms such as u′v′w′) in the equations. A new equation for this 3rd order term will contain even higher and complex terms that we don’t know what to do with. The fundamental problem here is that for any finite set of equations, the description of turbulence is not closed. This is called turbulence closure problem.

The turbulence closure problem is solved by retaining only a finite set of unknowns (up to a particular order) in the equations and the rest are parameterized (approximated) using known quantities and parameters. Such approximations, known as closure approximations/ schemes, are essential in order to have a closed set of equations to study the BL. Closure schemes are commonly named based on the order of statistical moments representing turbulent fluxes are retained in the equations, as well as based on whether approximations are carried out using local gradients or information from the whole model domain. As an example, first order local closure scheme will retain 1st statistical moments such as u, v, w and θ in the equations, and all 2nd moments u’w’, w’ θ’ are approximated using local gradients of u, v, w, θ.

For example, the zonal component of vertical momentum flux is approximated as:

6. Prandtl’s mixing length theory

According to Kinetic theory of gases, momentum and other properties are exchanged when molecules collide with each other, and molecular viscosity is expressed as a product of mean molecular velocity and mean free path length. Prandtl hypothesized a similar mechanism for turbulent transfer that eddies/ parcels broke away from the main body of the fluid will carry the mean properties of its original level until it travels a characteristic distance called the ‘mixing length’ (analogous to mean free path), before they mix with the new environment. Prandtl’s hypothesis is an attempt to specify eddy diffusion coefficient (K), used for approximating vertical fluxes in terms of vertical gradients, as a function of geometry and flow parameters, assuming that turbulent mixing is analogous to molecular diffusion.

In order for the parcel to move upwards a distance z’, it must have had some vertical velocity w’.

Prandtl assumed that in turbulent flow, velocity fluctuations in all directions are of the same order of magnitude. i.e. w’ is proportional to u’.

and expressed in m2 s-2. TKE is generated by buoyant thermals as well as by mechanically generated eddies. The diurnal evolution of TKE over land will have substantial variation with time. The values are usually small during night and early morning hours, followed by rapid increase to reach maximum in the early afternoon. Typically, TKE values vary in the range 0.01 – 1 m2s-2 close to the surface during a diurnal period. Other interesting facts about TKE are as follows:

• Large variation in TKE observed with time and height in the BL.
• Vertical profile of TKE may show maximum around z/zi = 0.3 when free convection dominates; due to contribution from energetic thermals at that height. zi is mixed layer height.
• In presence of strong wind shear, the maximum in TKE may occur very close to the surface followed by rapid decrease and nearly constant or slight decrease in the mixed layer.
• TKE is suppressed or lost by layers of air that are becoming more stable and is also dissipated into heat by the effects of molecular viscosity.

TKE is an important variable used to study turbulence and it’s evolution in the boundary layer. It is a measure of the intensity of turbulence and directly related to the momentum, heat and moisture transport through the boundary layer.

7.1. TKE budget

Turbulent kinetic energy budget is used to quantify the sources and sinks, and how turbulence/TKE varies with time. TKE is produced by mechanical (or shear) and buoyancy and destroyed/ suppressed by buoyancy suppression and molecular dissipation. TKE may also be transported into (a gain) or out (a loss) of a region/ layer. The tendency of TKE (increase or decrease with time) will tell whether the boundary layer is becoming less turbulent with time or becoming more turbulent with time. The role of different turbulence processes involved can be understood by looking into the TKE budget equation, which is a sum of production/loss terms and transport terms.

Choosing the coordinate system aligned with the mean wind (mean wind along x –direction), and neglecting subsidence, the TKE budget equation can be expressed as:

Term 1: Local storage or tendency term; Term 2: advection of TKE by mean flow;

Term 3: Buoyancy production or consumption term; Term 4: Mechanical/shear production;

Term 5: turbulent transport by eddies; Term 6: Pressure correlation term – how TKE is redistributed and drained out of PBL by pressure perturbations, often gravity waves; Term 7: Viscous dissipation term – conversion of TKE to heat

Term 1: Storage

• TKE values are usually small during night and early morning hours, followed by rapid increase to reach maximum in the early afternoon: net storage of TKE, positive storage term. The TKE tendency (de/dt) may vary between 5 x 10-5 m2s-3 to 5 x 10-3 m2s-3.
• During late afternoon and evening, decrease of TKE occurs due to dissipation and other losses, and storage/tendency term is usually negative. Diurnal variations of TKE over ocean are usually very small.

Term 2: Advection

• When averaged over a large horizontal area (e.g. 10 km x 10 km area or larger), one can assume that there is little horizontal variation of TKE and therefore, the advection term can be considered as negligible.
• Advection contribution will be significant if there are large changes in surface characteristics, and horizontal scale of interest is small.

Term 3: Buoyant production/consumption

This term represents vertical flux of virtual potential temperature and related to the generation or destruction of TKE by buoyancy.

Production of TKE

• Convective turbulence is caused by buoyancy forces that result from the heating of the surface. Parcel/thermal motions will have w′ and θ′ positively correlated during day time, positive heat flux and contributes positively to TKE.
• The buoyancy flux is positive and large near the ground and decreases approximately linearly with height up to bottom 2/3 rd of the mixed layer. This term represents the effect of thermals on the generation of TKE, hence, will be larger on sunny days, and weaker on cloudy days.
• Contribution of this term can be as large as 1 x 10-2 m2 s-3 near the ground. Buoyant production mostly generates turbulence in the vertical.

Consumption of TKE

• During statically stable conditions, air parcel displaced vertically by turbulence will be pushed back by buoyancy forces. Thus, static stability tend to suppress/ consume TKE.
• Parcel/thermal motions will have w′ and θ′ negatively correlated, and results in negative contribution to TKE budget.
• Buoyant consumption of TKE occur in SBL at night over land and during warm air advection over cold ground surface.

Term 4: Mechanical production

• Mechanical production term represents conversion of energy between mean flow and turbulent fluctuations and will be positive always.
• Mechanical turbulence is generated by the frictional retardation of the wind by the surface and roughness elements, i.e., due to wind shear.
• Mechanical production takes place during all conditions: statically stable, neutral and unstable BL conditions. Mechanical turbulence is the sole source of turbulence during neutral and stable conditions.
• In a statically stable atmosphere, turbulence can exist only if mechanical production is large enough to overcome damping effects of stability (buoyant suppression) and viscous dissipation.
• Mechanical production is usually large and positive near the ground.
• During calm wind and sunny conditions, buoyant production is dominant, i.e., buoyancy >> shear production, known as free convection.
• If shear production exceeds buoyant production (highly windy day), the associated convection is called forced convection. Forced convection would be more common with a synoptic-scale low than a high.

Term 5: Turbulent transport of TKE

• This term represents the vertical transport of TKE, i.e., vertical turbulent flux of TKE.
• It does not create or destroy TKE, but transports/redistributes it vertically before it is dissipated.
• This term also represents the flux divergence (or convergence) of TKE for a layer since it depends on the vertical gradient of vertical flux. For a given layer, if more flux is entering the layer than leaving it, there is a net convergence of the vertical flux, and therefore, the TKE of the layer will increase.
• Integrated over the whole depth of the BL, transport term becomes zero.
• Convective condition profiles may have a maximum transport around z/zi = 0.3 to 0.5

Term 6: Pressure correlation term

• This term is hard to measure since p’ is often quite small (~ 0.05 mb in stable BL and ~ 0.01 mb in CBL), therefore the w’p’ term is usually in the noise level. Usually estimated as a residual of the TKE budget equation, which will contain pressure correlation term + errors in other terms.
• Waves in the boundary layer are thought to contribute to this term. For a vertically propagating internal gravity wave, TKE may be lost from BL top.
• Contribution from this term is less than 10 % of the total rate of TKE dissipation.
• The pressure correlation term redistributes TKE in the BL as well as drains energy out of BL.

Term 7: Dissipation

• The dissipation term accounts for the smallest eddies giving up their energy in the form of viscous heating.
• Energy cascade – energy is transferred from the large eddies to the smaller ones, and eventually into heat at the molecular level
• The largest dissipation rates are near the ground (This is also where the buoyancy and shear source terms are largest), near constant within the ML and decreases rapidly above the mixed layer.

8. Stability and turbulent nature of BL

It is important to understand how turbulent mixing is affected by the static stability of the atmosphere. During day time, surface is heated and the boundary layer becomes unstable resulting in generation of convective plumes and turbulent mixing. In clear nights, surface is cooled and the boundary layer becomes stable, suppression of vertical movement of air parcels take place resulting in inhibition of turbulent mixing. Statically unstable flow is always dynamically unstable, whereas statically stable flow can be dynamically unstable (turbulent) or laminar. So it is important to infer: if we have a laminar flow in a statically stable BL, will it remain laminar, or will it become turbulent ?

By comparing the relative magnitudes of the shear production and buoyant terms in the TKE budget equation, and considering the sign of buoyancy term, we can infer the stability conditions, and when the statically stable flow might become dynamically unstable, and vice versa. A widely used approach in boundary layer research is based on gradient Richardson number (Ri).

Considering that w’θ’v is proportional to virtual potential temperature gradient and u’w’ is proportional to gradient in wind in the vertical, Ri is given by the ratio of buoyancy and shear terms in the TKE budget equation.

Here it is further assumed that mean flow is aligned along the x direction.

Ri is usually negative during daytime corresponding to unstable conditions and positive during stable nighttime conditions. The stable conditions can be laminar or turbulent, and their transitions can be inferred from the variation of the magnitude of Ri.

• Initial laminar flow: become unstable and onset of turbulence when Ri becomes smaller than the critical Richardson number Rc, i.e. when Ri < Rc.
• Initial turbulent flow: become laminar when Ri become greater than RT, where RT refers to termination of turbulence, i.e. when Ri > RT.
• Richardson number of non-turbulent flow must become less than Rc to initiate turbulence, but once turbulent, turbulence will continue until Ri is above RT.
• Studies have shown that Rc values are in the range 0.21 to 0.25 and RT = 1.0

9. The Ekman layer

The layer extending from the top of the surface layer to the top of the PBL, i.e. the outer layer, during neutral conditions is known as Ekman layer. This layer is characterized by turning of wind with height as the effect of friction diminishes and the wind approaches its geostrophic value at the top of BL.

The momentum equations in the BL can be modified to describe the turning of wind in the boundary layer. Applying first order local closure (K-theory) to the u and v components of momentum equations in BL (as discussed in section 5), and solving for u and v gives: turbulent transfer coefficient for momentum. , where f is the Coriolis parameter and Km is the

These equations describe the variation of wind speed and direction in the BL. When plotted on a hodograph, we obtain what is called the Ekman Spiral. The hodograph (Fig. 5) shows that the velocity vectors trace out a spiral shape with height, which is called the Ekman Spiral. The top of Ekman layer is defined as the layer where v = 0, i.e., where wind is parallel to and nearly equal to the geostrophic

wind ug, i.e., where the spiral crosses the u-axis. This occurs when , i.e. when . This depth, where wind becomes parallel to geostrophic wind is called Ekman layer depth given by The ideal Ekman layer discussed here is rarely observed in the atmospheric BL. The main reasons are:

Momentum flux is not proportional to the gradient of wind speed throughout the BL as assumed in the Ekman model.

The value of Km varies drastically with height near the ground, whereas in the model it is assumed constant throughout the BL.

11. Summary

• Turbulent motions, characterized by random irregular motions in the flow, are responsible for the vertical exchange of energy, momentum and mass in the atmospheric boundary layer. 
• Atmospheric turbulence consists of wide range of scales of motion, with energy spectra indicating 3 regions corresponding to energy producing large eddies, intermediate eddies in the inertial sub-range involved in energy transfer and very small eddies in the dissipation region.
• Prandtl’s mixing length theory provides a mechanism to estimate eddy transfer coefficients during near neutral conditions.
• Conservation equations for the atmospheric boundary layer incorporates turbulent flux gradient terms with significantly large magnitude. 
• Closure relations of different types are used to approximate the flux terms in terms of known variable to make a closed set of equations.
• Turbulent kinetic energy budget provides understanding on sources, sinks and transport of turbulence in the atmospheric boundary layer.
• Gradient Richardson number can be used to infer stability of the surface layer and transition between laminar and turbulent motions.
• The Ekman layer solutions can be used to represent turning of wind with height above the surface layer, in neutral boundary layer conditions.