Surface Fluxes of Momentum, Heat, and Water Vapor
65
last approach the fluxes (covariances) are not determined directly but are derived from turbulence measurements and use of variance budget equations that relate surface fluxes to mean
dissipation rates. Therefore, we begin the discussion of this method with an examination of
the budget equations for turbulent kinetic energy and temperature variance (with temperature
serving as a proxy for any other scalar).
3.1.3 Turbulent kinetic energy
The mean turbulent kinetic energy (TKE) dissipation rate (c:) is related to the production rate
of TKE and hence the surface fluxes through the TKE budget equation. The TKE budget
equation is obtained by multiplying the momentum equation for the a(= 1,2,3) direction,
by u"" time averaging all terms and subtracting the equation governing the kinetic energy
of the mean flow (Tennekes and Lumley, 1972, p63). For a stationary flow with horizontal
homogeneity we write the budget equation for the time averaged TKE per unit mass with the
Boussinesq approximation as
8U
g < we >
- < uw > - + =----::--8z
e
18
2
8z
18
---:=:--=c:
p
8z
(3.7)
where p is the fluctuating component of the pressure, and e is twice the TKE (i.e. TKE =~ =
u 2 +v;+w2). The first term in (3.7) represents the mechanical (or shear) production of TKE by
interaction of the Reynolds stress with the mean velocity gradient. The second term represents
the TKE production by buoyancy, the third the divergence of vertical turbulent flux of TKE
and the fourth the transfer term for TKE due to pressure-velocity interaction. The latter two
terms on the left hand side of (3.7) are generally called the transport terms. This budget is
non-dimensionalized by u~/kz to arrive at
(3.8)
where
• L
u;
(3.9)
and F and P are written for 1 8 and 1 8 respectively.
2
8z
p
8z '
It is often assumed that total production (mechanical plus buoyant) equals dissipation (Deacon,
1988), regardless of the state of atmospheric stratification and position in the boundary layer,
i.e.
(3.10)
where
In the Kansas experiments, Wyngaard and Cote (1971; abbreviated here as WC71 ) made
measurements of the production, turbulent transport and dissipation of TKE for near neutral
and moderately unstable stratification. Their results showed dissipation exceeding production
65
last approach the fluxes (covariances) are not determined directly but are derived from turbulence measurements and use of variance budget equations that relate surface fluxes to mean
dissipation rates. Therefore, we begin the discussion of this method with an examination of
the budget equations for turbulent kinetic energy and temperature variance (with temperature
serving as a proxy for any other scalar).
3.1.3 Turbulent kinetic energy
The mean turbulent kinetic energy (TKE) dissipation rate (c:) is related to the production rate
of TKE and hence the surface fluxes through the TKE budget equation. The TKE budget
equation is obtained by multiplying the momentum equation for the a(= 1,2,3) direction,
by u"" time averaging all terms and subtracting the equation governing the kinetic energy
of the mean flow (Tennekes and Lumley, 1972, p63). For a stationary flow with horizontal
homogeneity we write the budget equation for the time averaged TKE per unit mass with the
Boussinesq approximation as
8U
g < we >
- < uw > - + =----::--8z
e
18
2
8z
18
---:=:--=c:
p
8z
(3.7)
where p is the fluctuating component of the pressure, and e is twice the TKE (i.e. TKE =~ =
u 2 +v;+w2). The first term in (3.7) represents the mechanical (or shear) production of TKE by
interaction of the Reynolds stress with the mean velocity gradient. The second term represents
the TKE production by buoyancy, the third the divergence of vertical turbulent flux of TKE
and the fourth the transfer term for TKE due to pressure-velocity interaction. The latter two
terms on the left hand side of (3.7) are generally called the transport terms. This budget is
non-dimensionalized by u~/kz to arrive at
(3.8)
where
• L
u;
(3.9)
and F and P are written for 1 8
2
8z
p
8z '
It is often assumed that total production (mechanical plus buoyant) equals dissipation (Deacon,
1988), regardless of the state of atmospheric stratification and position in the boundary layer,
i.e.
(3.10)
where
In the Kansas experiments, Wyngaard and Cote (1971; abbreviated here as WC71 ) made
measurements of the production, turbulent transport and dissipation of TKE for near neutral
and moderately unstable stratification. Their results showed dissipation exceeding production