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Prognostic equations for turbulent variances

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Title: Prognostic equations for turbulent variances


1
  • Prognostic equations for turbulent variances
  • 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 (4.3)

2
Prognostic equation for momentum variance
  • Start with the complete equation
  • multiply by 2ui, rearrange ( )
    and average all equation after rearranging
  • Finally, being (turbulent continuity equation)

3
Prognostic equation for momentum variance
  • And we can write the general form for the
    prognostic equation for variance of wind speed
  • Further simplifications are then possible they
    will involve the terms of dissipation, pressure
    perturbations and Coriolis

4
Simplifications in the eq. of momentum variance
(1)
  • Dissipation it is possible, by following
    rewriting
  • to give
  • First term in the r.h.s. represents the molecular
    diffusion of velocity variance and contains the
    curvature of variance, and ranges in the BL
    between 10-11?10-7 m2s-3
  • Last term in r.h.s. is quite larger, ranging
    between 10-4?10-3 m2s-3 in the ML and 10-2 m2s-3
    in the SL
  • Viscous dissipation e is defined as
    is always loss term

5
Simplifications in the eq. of momentum variance
(2)
  • Pressure perturbations
  • it is possible to rewrite this term as
  • the last term is called pressure redistribution
    term the 3 spatial derivatives in the bracket
    term sum to 0 (turbulent continuity equation), so
    this term dont change total variance and can be
    neglected, even if it redistributes energy from
    more energetic to less energetic eddies

6
Simplifications in the eq. of momentum variance
(3)
  • The Coriolis term is identically zero for
    velocity variances, as it can be seen by
    performing the sums implied by the repeated
    indices (remember that Reynolds stress tensor is
    symmetric)
  • Kinetic energy is related to variance as is
    twice TKE (per unit mass)
  • Coriolis term effect is to redistribute energy
    from one horizontal direction to another, but it
    is normally neglected as its numerical value is
    about 3 order of magnitude smaller than other
    terms

7
Passages in momentum variance equation
8
Passages
9
Simplified velocity variance budget equation (1)
Before simplifications and rearranging
After rearranging original equation, we have
10
Simplified velocity variance budget equation (2)
  • After rearranging original equation, we have
  • Term I represents local storage of variance
  • Term II describes the advection of variance by
    the mean wind
  • Term III is a production or loss term, depending
    on whether buoyancy flux ltwqgt is positive or
    negative
  • Term IV is a production term momentum flux
    ltuiujgt is usually negative in the BL because
    wind momentum is lost downward to the ground
  • Term V is a turbulent transport term, describing
    how variance ltui2gt is moved around by the
    turbulent eddies uj
  • Term VI describes how variance is redistributed
    by pressure perturbations, and is associated with
    air oscillations (gravity waves)
  • Term VII represents viscous dissipation of
    velocity variance

I II III
IV V
VI VII
11
Prognostic equations for each individual wind
component
  • If we put i1,2,3 in the general prognostic
    equation, we obtain the equation for the
    individual component u,v,w (remember to reinsert
    the return-to-isotropy terms (non null for each
    individual component)
  • Terms I-VII have same meaning as above term VIII
    represents pressure redistribution, associated
    with return-to-isotropy term

I II III
IV V VI
VIII VII
12
Vertical velocity variance variation (1)
Vertical velocity variance during daytime is
small near the surface, then increases to a
maximum about a third of the distance from the
ground to the top of ML and then decreases with
height. This is related to the vertical
acceleration experienced by thermals during their
initial rise, reduced by dilution with
environmental air, by drag, and by the warming
and stabilizing of the environmental near the top
of ML. In cloud-free conditions, glinder pilots
and birds would expect to find maximum lift at
z/zi0.3
13
Vertical velocity variance variation (2)
At night, turbulence rapily decreases over the
RL, leaving a much thinner layer of turbulent air
near the ground.
14
Velocity variance
The depth of the turbulent nocturnal SBL is often
relatively small (h?200m). In statically neutral
conditions the variances also decrease with
height from large values at surface.
15
Horizontal velocity variance
The horizontal components of velocity variance
are often largest near the ground during the day,
associated with the strong wind shears in the SL.
The horizontal variance is roughly constant
throughout the ML, but decreases with height
above the ML top.
16
Horizontal eddy kinetic energy
At night, the horizontal variance decreases
rapidly with height to near zero at the top of
the SBL. This shape is similar to that of the
vertical velocity variance
17
Moisture variance
  • Start with equation for turbulent part of water
    vapour
  • Multiply for 2q, use Reynolds rules
    (2q?q/?t?q2/?t), average and apply averaging
    rules, and add average turbulent continuity
    equation (0)
  • Last term can again be splitted into 2 parts, one
    small enough to be neglected remaining term eq
    is twice the molecular dissipation term

18
Prognostic equation for moisture variance
  • The final equation can be written as
  • Term I represents local storage of humidity
    variance
  • Term II describes the advection of humidity
    variance by the mean wind
  • Term IV is a production term, associated wit
    turbulent motions occurring within a mean
    moisture gradient
  • Term V is a turbulent transport of humidity
    variance
  • Term VII is molecular dissipation

I II IV
V VII
19
Passages for moisture variance equation
20
Specific humidity variance
Humidity variance is small near ground, because
thermals have near the same humidity as their
environment. At the top of the ML, however, drier
air from aloft is being entrained down between
the moist thermals, creating large humidity
variances. Part of this variance might be
associated with the excitation of
gravity/buoyancy waves by the penetrative
convection
21
Specific humidity variance
Production terms balance loss terms in the
budget, assuming a steady state situation when
storage and mean advection are neglected. Notice
than transport terms (found to be as the
residual) are positive in the bottom half of the
ML, but are negative in the top half. The
integrated effects of these terms are zero. Such
is the case for most transport terms they
merely move moisture variance from one part of
the ML (where there is excess prodiction) to
another part (where there is excess dissipation),
leaving zero net effect when averaged over the
whole ML
22
Heat (potential temperature variance)
  • Similarly to moisture equation start with
    equation for turbulent temperature
  • Multiply for 2q, use Reynolds rules
    (2q?q/?t?q2/?t), average and apply averaging
    rules, and add average turbulent continuity
    equation (0)
  • Last term (VIII) is the radiation destruction
    term (sometimes indicated as eR), while other
    terms are physically analogous to moisture
    equation corresponing ones

I II IV
V VII VIII
23
Passages for thermodynamic equation
24
Virtual potential temperature variance (1)
The temperature variance at the top of the ML is
similar to humidity variance, because of the
contrast between warmer entrained air and the
cooler overshooting thermals. Gravity waves may
also contribute to the variance. There is a
greater difference near the bottom of the ML,
however, because warm thermals in a cooler
environment enhance the magnitude of the variance
there.
25
Virtual potential temperature variance (2)
At night, the largest temperature fluctuations
are near the ground in the NBL, with weaker
sporadic turbulence in the RL aloft
26
Virtual potential temperature variance (3)
The radiation destruction term is small, but
definitly non-zero. The dissipation is largest
near ground, as is the turbulent transport of
temperature variance. The storage and advection
terms are not shown here.
27
Virtual potential temperature variance (4)
At night, the behavior of the corresponding terms
is quite different particularly in the lower
layers
28
Scalar quantity variance
  • Similarly to water vapour equation
  • Multiply for 2c, use Reynolds rules
    (2c?c/?t?c2/?t), average and apply averaging
    rules, and add average turbulent continuity
    equation (0)
  • Last term can again be splitted into 2 parts, one
    small enough to be neglected remaining term ec
    is twice the molecular dissipation term

I II IV
V VII
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