Geoscience Reference
In-Depth Information
Fig. 3.8
Impact of averaging time on turbulent heat flux calculated from the covariance of devia-
tory temperature and vertical velocity for a 6-h period. Numbers next to the symbols indicate the
numberofrealizationsineachaverage. Errorbarsindicatetheconfidenceintervalforeachaverage,
derived via the bootstrap method as described in the text
3.3 Turbulent Kinetic Energy Equation
One of the most useful tools for understanding how turbulence works in natural
flows is the
turbulent kinetic energy
(TKE) equation. Starting from the Boussinesq
formoftheEulerequation(2.16)
∂
u
i
∂
u
j
∂
u
i
1
ρ
∂
p
ρ
ρ
δ
i
3
+
∂τ
ji
g
t
+
x
j
+
2
ε
ijk
Ω
j
u
k
=
−
x
i
−
(3.1)
∂
∂
∂
x
j
where
u
i
=
u
i
is the instantaneous velocity,
p
is pressure after removal of the
hydrostaticpart,and
U
i
+
istheearth'srotationvector;
2
anequationforkineticenergy
(per unit mass) of the flow is obtained by multiplying (3.1) by
u
i
/
Ω
2. If the corre-
spondingequationforthekineticenergyofthemeanflow(
U
)issubtractedfromthe
instantaneousequation,the result is (after eliminatingsome termsbased onscaling
2
In these equations, repeated indices imply summation as before, the Kronecker
δ
ij
is 1 for
i
=
j
and0otherwise, andthealternatingtensoris
ε
123
=
ε
231
=
ε
312
=
−
ε
321
=
−
ε
213
=
−
ε
132
=
1and
0 otherwise. Cartesian tensor notation issummarized by Hinze (1975, Appendix).
