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Turbulent Kinetic Energy TKE

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TKE (Turbulence Kinetic Energy) is a measure of the intensity of turbulence. ... of vertical kinetic energy into horizontal directions is return-to-isotropy term ... – PowerPoint PPT presentation

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Title: 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
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