Geoscience Reference
In-Depth Information
argument)theTKEequation(see, e.g.,Hinze1975;TennekesandLumley1972):
u
i
u
i
2
u
i
u
i
2
+
u
i
u
j
∂
x
j
u
i
u
i
u
j
(3.2)
∂
∂
U
j
∂
∂
U
i
1
2
∂
+
x
j
+
t
x
j
∂
∂
∂
∂
x
i
p
u
i
−
ρ
u
i
δ
i
3
−
ν
u
i
u
j
∂
u
j
∂
x
j
+
∂
1
ρ
∂
∂
g
ρ
=
−
∂
x
i
x
i
u
i
u
i
where
(thetraceof theReynoldsstress tensor)is twice theturbulentkinetic
energyperunitmass.
Equation (3.2) is greatly simplified if the flow is steady and horizontally homo-
geneous(i.e.,horizontalgradientsnegligiblecomparedwith vertical)with nomean
verticalmotion:
P
S
←−−−−−−−−−−→
P
b
←−−−→
D
←−−−−−−−−−−→
ε
←−−−−−−−−−−→
∂
u
i
∂
x
i
∂
u
j
x
i
(3.3)
ρ
ρ
w
=
∂
z
(
+
ρ
)+
ν
u
i
u
i
w
p
w
∂
u
j
∂
u
w
∂
U
∂
v
w
∂
V
∂
g
−
z
−
z
−
x
j
+
2
∂
In this case, the TKE balance comprises four terms. The first term on the left is
the production rate of TKE by the interaction of turbulent stress with mean shear:
P
S
=
τ
·
∂
z
.Inshearflowsasconsideredhere,itispositive,meaningthatthein-
teraction of shear with Reynoldsstress producesturbulence.A correspondingterm
with opposite sign occursin the mean flow kinetic energy equation,which demon-
stratesmathematicallyhowturbulencedrainskineticenergyfromthemeanflow.
ThesecondtermontheleftistheproductionofTKEbybuoyancy(gravitational)
forcesin thefluid,andmaybeexpressedintermsof
buoyancy flux
:
w
b
=
U
/
∂
ρ
w
=
−
g
ρ
P
b
(3.4)
If TKE buoyancy production is positive (buoyancy flux negative), the flux of pos-
itive density variations is downward and energy is added to turbulence by gravity.
If buoyancy production is negative (buoyancy flux positive), TKE is expended in
moving denser fluid upward.
P
b
thus represents the conversion between TKE and
potential energy of the fluid, and often has a profound impact on the scales of tur-
bulenceas exploredinmoredetailinChapter5.
The term labeled
D
is the divergence of a combination of vertical flux of TKE
andof the covarianceofverticalvelocitywith turbulentpressurefluctuationsin the
flow. Both parts are difficult to measure accurately, and in the atmospheric surface
layer(thefirstfewdekameters),scalingargumentsindicatethat
D
isoftennegligibly
smallcomparedwiththeotherthreeterms.Fortheouterpartoftheboundarylayer,
thisisnotnecessarilyvalid,andindeed,longaveragesofTKEfluxatmultiplelevels
during the SHEBA project indicated the TKE flux divergence
u
i
u
i
w
/
∂
(
∂
z
)
to
beimportantintheTKEbalance(McPhee2004).
Thelastterm,
,whichinvolvesgradientsoversmalldistancesinallcomponents
of the flow, is the dissipation of TKE into internal energy (heat) of the fluid. It is
ε
