非线性动力学（英文） (12).pdf

11 Lorenz equations
In this lecture we derive the Lorenz equations, and study their behavior.
The equations were first derived by writing a severe, low-order truncation of
the equations of R-B convection.
One motivation was to demonstrate SIC for weather systems, and thus point
out the impossibility of accurate long-range predictions.
Our derivation emphasizes a simple physical setting to which the Lorenz
equations apply, rather than the mathematics of the low-order truncation.
See Strogatz, Ch. 9, for a slightly different view. This lecture derives from Tritton, Physical Fluid
Dynamics, 2nd ed. The derivation is originally due to Malkus and Howard.
Physical problem and parameterization
We consider convection in a vertical loop or torus, ., an empty circular
tube:
cold
g
hot
We expect the following possible flows:
Stable pure conduction (no fluid motion)
•
Steady circulation
•
Instabilities (unsteady circulation)
•
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The precise setup of the loop:
T0−T1 (external)
T0−T2
z
)
(T0−T3) (T0+T3 g
q
a φ
T 0+T 2
T0+T1 (external)
θ= position round the loop.
External temperature TE varies linearly with height:
T = T T z/a = T + T cos θ(24)
E 0 − 1 0 1
Let a be the radius of the loop. Assume that the tube’s inner radius is much
smaller than a.
Quantities inside the tube are averaged cross-sectionally:
velocity = q = q(θ, t)
temperature = T = T (θ, t) (inside the loop)
As in the Rayleigh-B´enard problem, we employ the Boussinesq approximation
(here, roughly like pressiblity) and therefore assume
ωδ
= 0.
ωt
Thus mass conservation, which would give ψu in the full problem, here
·
gives
ωq
= 0. (25)
ωθ
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Thus motions inside the loop are equivalent to a kind of solid-body rotation,
such that
q = q(t).
The temperature T (θ) could in reality vary with plexity. Here we
assume it depends on only two parameters, T2 and T3, such that
T T = T