Prognostic equations for turbulent variances

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• 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)

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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)

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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

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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

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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

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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

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Passages in momentum variance equation
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Passages
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Simplified velocity variance budget equation (1)
Before simplifications and rearranging
After rearranging original equation, we have
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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
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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
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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
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Vertical velocity variance variation (2)
At night, turbulence rapily decreases over the
RL, leaving a much thinner layer of turbulent air
near the ground.
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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.
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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.
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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
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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

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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
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Passages for moisture variance equation
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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
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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
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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
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Passages for thermodynamic equation
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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.
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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
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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.
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Virtual potential temperature variance (4)
At night, the behavior of the corresponding terms
is quite different particularly in the lower
layers
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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