Boundary%20Layer%20Meteorology%20Lecture%204

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Boundary Layer Meteorology Lecture 4

• Turbulent Fluxes
• Energy Cascades
• Turbulence closures
• TKE Budgets

2
Turbulent Fluxes

• Terminology
• Stationary Turbulence is constant is independent
  of translation along the time axis.
• Homogeneous Turbulence is independent of
  translation along any spatial axis.
• Isotropic Turbulence is statistically independent
  of rotation, reflection or translation

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Energy Cascades

• Kolmogorov Theory
• Existence of equilibrium range of scales of
  motion in which the average properties are
  determined uniquely by the fluids viscosity and
  the total dissipation. ?k
  ???????????v?????????
• Existence of an inertial subrange within the
  equilibrium range, but removed from the scales
  where viscous forces are important, where only
  inertial transfer of energy is important. 1/ls
  ltlt ? ltlt 1/?k

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Turbulence Closures

• Equation 2.42 has unknowns like
  . How do we solve for it?
• Note that the problem arises from Reynolds
  Averaging. Why do we need Reynolds averaging?
  Because we dont have enough computing power to
  span the whole inertial subrange, bridging the
  scales where viscosity matters to the scales we
  care about (10s of meters to a few kilometers in
  the vertical). As Garratt points out, wed need
  a trillion grid points to do this explicitly.
  Can we fudge it?

5
First order closures

• Can be as simple as assuming us K ds/dx, with
  constant K
• Or as complicated as prescribing a vertical and
  time dependence of K (which will also vary
  depending on whats being transported), where K
  might depend on the local shear and on the
  Richardson number.

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One-and-a-half oder closure

• In this case we acknowedge that an eddy
  diffusivity (K) ought to depend on the kinetic
  energy of the eddy part of the flow so we
  should maybe predict the TKE for each location.
  Further simplifications are introduced to make
  this possible (see chapter 8).

7
Second-order closure

• Maybe if a crude assumption for nth order
  moments predicts (n-1)th moments adequately,
  perhaps a similar assumption for (n1)th moments
  will predict nth moments just as well (Lumley
  Khajeh-Nouri, 1974, quoted by Garratt).
• uiujuk terms usually parameterized by assuming
  down-gradient diffusion of uiuj terms.
• Dissipation usually parameterized in terms of TKE
  and a large-eddy length scale (probably at least
  a little arbitrary).

8
TKE budget equation

• De/Dt S B T DS shear production B
  bouyancy flux T transport pressure work D
  dissipation
• See Bretherton Notes 3.