Turbulent Kinetic Energy TKE

1

• Turbulent Kinetic Energy (TKE)

• Textbooks and web sites references for this
  lecture
• Roland B. Stull, An Introduction to Boundary
  Layer Meteorology, Kluwer Academic Publishers,
  1989, ISBN 90-277-2769-4 ( 5.1-5.5)

2
TKE

• TKE (Turbulence Kinetic Energy) is a measure of
  the intensity of turbulence.
• It is directly related to the momentum, heat, and
  moisture transport through the boudary layer
• The individual terms in the TKE budget equation
  describe physical processes that generate
  turbulence
• The relative balance of these processes
  determines the ability of the flow to mantain
  turbulence or become turbulent, and thus
  indicates flow stability
• Some important dimensionless groups and scaling
  parameters are also based on terms in TKE
  equation

3
TKE

• TKE has been defined as
• ltegt 0.5 ( ltugt2 ltvgt2 ltwgt2 ) 0.5 ltuigt2
• This is sum of velocity variances devided by two
• We can directly take the variance equation
• And divide by two and substitute ltegt

I II III
IV V
VI VII
4
Meaning of the terms

• Term I represents local storage or tendency of
  TKE
• Term II describes the advection of TKE by the
  mean wind
• Term III is the buoyant production or consumption
  term it is gain or loss depending on whether
  heat flux ltuiqgt is positive (during daytime
  over land) or negative (at night over land)
• Term IV is a mechanical or shear production/loss
  term the momentum flux ltuiujgt is usually of
  opposite sign from the mean wind shear, because
  the momentum of the wind is usually lost downward
  to the ground (friction) thus, term IV results
  in a positive contribution to TKE when multiplied
  by a negative sign
• Term V represents the turbulent transport of TKE,
  describing how TKE is moved around by the
  turbulent eddies uj
• Term VI is a pressure correlation term that
  describes how TKE is redistributed by pressure
  perturbations it is often associated with air
  oscillations (buoyancy or gravity waves)
• Term VII represents viscous dissipation of TKE
  that means conversion of TKE into heat

5
X-wind aligned subsidence-free equation

• By choosing coordinate system aligned with mean
  wind, assuming horizontal homogeneity and
  neglecting subsidence, we can have
• Turbulence is dissipative.
• Term VII is a loss term that always exists
  whenever TKE is non-zero
• Physically, this means that turbulence will tend
  to decrease and disappear with time, unless it
  can be generated locally (term I) or transported
  in by the mean (term II), turbulent (term III, IV
  and V) or pressure (term VI) processes
• Thus, TKE is not a conserved quantity
• BL can be turbulent only if they are specific
  physical processes generating turbulence

I III IV
V VI VII
6
TKE storage

• These diurnal variations represent net storage of
  TKE into air in particular non-turbulent FA air
  just above the ML top, is entrained and
  incorporated in the ML
• Over land, storage term can vary in the range 5
  10-5 m2 s-3 for surface-layer air over 6 h
  interval, to about 5 10-3 m2 s-3 for FA air over
  15 min in early morning
• During late afternoon, there is spin down
  (decrease with time) because TKE dissipation and
  losses exceed TKE production. The storage term is
  thus negative during this transition phase

• Magnitude of TKE varies with time at
• any height within diurnal cycle there
• are dramatic increase and decrease
• of TKE

7
TKE storage (cont)

• Vertical profile of TKE can sometimes increase to
  a maximum at a height of about z/zi?0.3 when free
  convection dominates
• When strong winds are present, TKE can be nearly
  constant with height within BL, or might decrease
  slightly with height
• At night, TKE often decreases very quickly with
  height, having maximum value near surface
• Over oceans, where diurnal cycle is not too
  evident, storage term is so small that it can be
  neglected ? intensity of TKE do not change too
  much with time

8
TKE advection

• When averaged over an horizontal area larger than
  about 10 Km, it is often assumed that there is
  little horizontal variation in TKE, thereby
  making this term negligible (good assumption for
  most land surfaces)
• On smaller scale, this term can be relevant,
  particularly when surface characteristics are
  very different (sea-land, )
• Over ocean, this term probably is negligible also
  at smaller scales

9
TKE buoyant production/consumption

• The most important part of the buoyancy term is
  the flux of virtual potential temperature,
  ltwqvgt
• This flux is positive and decreases roughly
  linearly with height within the bottom 2/3 of
  convective ML
• Near ground, term III is large and positive,
  corresponding to a large generation rate of
  turbulence whenever the underlying surface is
  warmer than air
• When positive, this term represents the effects
  of thermals in the ML. Active thermal convection
  is associated with large values of this term
  (10-2 m2s-3 at ground)
• This term is associated with sunny days over
  land, or cold air advection over warm surface
• During cloudy days, it can be much smaller

10
TKE buoyant production/consumption (cont)

• If convective BL is capped by actively growing
  cumulus clouds, the positive buoyancy within the
  cloud can contribute to TKE production (term III)
• Between this cloud layer contribution and the
  contribution near bottom of the subcloud layer,
  there can be a region near cloud base in which
  the air is statically stable and the buoyancy
  term is therefore negative

11
Dimensionless convective form of TKE equation

• Because term III is so important on days of free
  convection, this term is used to normalize all
  other terms
• Term III can be rewritten as
  at surface then, dividing all TKE equation
• By definition, term III in normalized equation is
  unity at surface
• Buoyancy term acts only on vertical component of
  TKE hence, this production term is anisotropic
  the term encharged to move some of vertical
  kinetic energy into horizontal directions is
  return-to-isotropy term

I III
IV V VI
VII
12
TKE consumption

• In statically stable conditions, an air parcel
  displaced vertically by turbulence would
  experience a buoyancy force pushing it back
  towards its starting height
• Static stability thereby tends to suppress or
  consume TKE, and is associated with negative
  values of term III
• Such conditions are present in SBL at night over
  land, or anytime surface is colder than overlying
  air
• Same type of consumption can occur at top of ML,
  when warmer air entrained downward by turbulence
  opposes the descent because of its buoyancy in
  this case, buoyancy term has negative values

13
TKE mechanical (shear) production

• When there is a turbulent momentum flux in the
  presence of a mean wind shear, the interaction
  between the two tends to generate more turbulence
• Even though a negative sign preceeds term IV, the
  momentum flux is usually of opposite sign from
  the mean shear, resulting in production, not
  loss, of turbulence
• The greatest wind shear magnitude and shear
  production occurs near surface
• Wind speed varies little with height in ML above
  SL, resulting in near zero shear and shear
  production of turbulence
• A smaller maximum of shear production sometimes
  occurs at top of ML because of entrainment, where
  subgeostrophic wind speed recover to their
  geostrophic values

14
TKE mechanical (shear) production (cont)

• Magnitudes of shear production term in SL are
  greatest on a windy day and small on a calm day
• In synoptic scale cyclones there are forced
  convection conditions
• On many days both shear and buoyancy contributes
  to product turbulence
• At night over land, or anytime ground is colder
  than air, shear term is often the only term
  generating turbulence as shear term is active
  only on a relatively small depth of air, this is
  reason for which NBL is usually thinner than ML
• Except on thunderstorms, shear of w is negligible
  in BL this means thatshear production is
  greatest in x and y components, i.e. in the
  horizontal (anisotropic)

15
TKE mechanical (shear) production (cont)

• Then both buoyancy and shear produce
  anisotropically turbulence buoyancy in the
  vertical and shear in the horizontal
• The structures due to free convection are
  predominantly vertical, while the forced
  convection eddies are sheared into a much more
  horizontal or slanting orientation, with a more
  chaotic appearence

16
TKE turbulent transport

• Quantity ltwegt represents vertical turbulent flux
  of TKE change of flux with height is more
  important that magnitude of flux
• Term V is a flux divergence term if in a layer
  there is more flux entering than leaving, then
  magnitude of TKE increases
• On local scale, term V acts as either production
  or loss, depending on whether there is flux
  convergence or divergence
• When integrated over ML depth, hovewer, term V
  becomes identically zero, assuming at bottom and
  top boundaries that earth is not turbulent and
  also that there is negligible turbulence above
  top of ML

17
TKE turbulent transport (cont)

• Term V neither creates or remove TKE, it just
  moves or redistribute TKE from one location in
  the BL to another
• During daytime convective cases, maximum of ltwegt
  is located at z/zi0.3 to 0.5
• Below this maximum, there is more upward flux
  leaving the top of any one layer that enters from
  below, making a net divergence or loss of TKE
• Above this maximum there is a net convergence or
  production of TKE
• Net effect is that some of TKE produced near
  ground is transported up to top half of ML before
  it is dissipated
• Splitting vertical turbulent transport of TKE
  into transport of w2 and of (u2v2), we can
  see that transport of w2 dominates in the middle
  of ML, while transport of (u2v2), dominates
  near the surface

18
TKE pressure correlation turbulence

• Static pressure fluctuations are very difficult
  to measure in the atmosphere their magnitudes
  are of the order of 0.05 hPa in the SL and 0.01
  hPa in the ML
• This term is normally calculated as residual term
  from budget equation

19
TKE pressure correlation waves

• Gravity wave theory shows that ltwpgt is equal to
  upward flux of wave energy for a vertically
  propagating internal gravity wave within a
  statically stable environment
• This suggests that turbulent energy can be lost
  from the ML top in the form of gravity waves
  being excited by thermals penetrating the stable
  layer at top of ML
• Amount of energy lost may be of order of 10 of
  the total rate of TKE dissipation
• Pressure correlation term then acts not only to
  redistribute TKE within BL, but also in draining
  energy out of BL

20
TKE dissipation

• Molecular destruction of turbulent motions
  greatest for the smallest size eddies
• The more intense is small-scale turbulence, the
  greater the rate of dissipation
• Small-scale turbulence is, in turn, driven by the
  cascade of energy from the larger scales
• Daytime dissipation rates are often largest near
  the surface, and then become relatively constant
  with height in ML
• Above ML top, dissipation rate rapidly decreases
  near zero
• At night, both TKE and dissipation rate decrease
  very rapidly with height

21
TKE dissipation

• TKE is not conserved, then the greatest values of
  TKE and dissipation rates are frequently found
  where TKE production is largest, i.e. near the
  surface
• In a typical example of dissipation rate with
  time from night to day, we can see that at night,
  where only shear can produce turbulence,
  dissipation rate is very small because TKE is
  small, while after sunrise buoyant production of
  TKE greatly increases turbulence intensity and
  also dissipation rate grows.

22
MKE

• Term IV of TKE equation refers to
  production of TKE by interaction of turbulence
  with mean wind
• We can expect that the energy involved in this
  production of TKE will correspond to a loss of
  kinetic energy from the mean flow
• Let start with equation for mean wind, multiply
  for ltuigt

?
To obtain
23
MKE
I II
III IV V
VI X

• Term I represents storage of MKE
• Term II describes the advection of MKE by the
  mean wind
• Term III indicates that gravitational
  acceleration of vertical motions alter the MKE
• Term IV shows the effects of the Coriolis force
• Term V represents the production of MKE when
  pressure gradients accelerate the mean flow
• Term VI represents the molecular dissipation of
  mean motions motions
• Term X indicates the interaction between the mean
  flow and the turbulence

24
MKE and TKE

• Term X can be rewritten as
• Also, gravity term is simply gltwgt and Coriolis
  term, when summed over all indices, vanishes
• In this case, equation for MKE can be rewritten
  as follows and compared with TKE equation
• In both equations is present the term of
  interaction between the mean flow and the
  turbulence, with opposite signs energy that is
  mechanically produced as turbulence is lost from
  the mean flow, and vice versa