Abstract
Abstract
We consider the behaviour of a passive tracer θ governed by
∂
t
θ
+
u
⋅
∇
θ
=
Δ
θ
+
g
in two space dimensions with prescribed smooth random incompressible velocity
u
(
x
,
t
)
and source g(x). In 1959, Batchelor et al (J. Fluid Mech.
5 113) predicted that the tracer (power) spectrum should scale as
|
θ
k
|
2
∝
|
k
|
−
4
|
u
k
|
2
for
|
k
|
above some
κ
¯
(
u
)
, with different behaviour for
|
k
|
≲
κ
¯
(
u
)
predicted earlier by Obukhov and Corrsin. In this paper, we prove that the BHT scaling does indeed hold probabilistically for sufficiently large
|
k
|
, asymptotically up to controlled remainders, using only bounds on the smaller
|
k
|
component.