Long solitary internal waves in stable stratifications

W. B. Zimmerman and J. M. Rees: Long solitary waves 169
solitary internal waves. The focus on the Scorer parameter,
given the indication that when L 2 > 0, there is apparent
correlation with observed wavelengths, begs the question of
the mechanism of wavelength selection. Further, if there is
a wavelength selection for waveguides with L 2 > 0, is there
none for L 2 < 0? The theory developed in the next section is
motivated by these questions.
3 Model system
3.1 Equations and scaling
Consider a fluid layer bounded above and below by parallel
material surfaces. The upper material surface is constrained
to move at velocity U = γ H , where γ is the shear rate, and
is held at constant temperature T = 1. The lower is held at
T = 0 and at rest. As the model that is proposed is Galilean
invariant, an arbitrary uniform wind component could be included
without loss of generality. The fluid is inviscid and
nonconducting. All lengths are scaled by H , velocity by
U, and time by H/U. The flow is assumed to be twodimensional
and divergence free, thus expressible through
a streamfunction, (u, v) = ( )
y , − x . Scalar transport is
modelled as convection only:
T t + y T x − x T y = 0 (8)
A Boussinesq fluid is assumed, with density linearly related
to temperature. The dimensionless density is ρ = 1−βT ,
where β is the dimensionless group formed from the coefficient
of thermal expansion β ∗ , the unscaled temperature
difference between the upper and lower surfaces T , and
ρ 0 , the density of the fluid at the reference temperature of the
lower surface. Namely, β = β ∗ T /ρ 0 .
Vorticity ω = − ∇ 2 is transported by
0 = RiT x + [ yt + y xy − x yy
]y
+ [ xt + y xx − x xy
]x
(9)
which is the curl of the Boussinesq momentum equation. The
Richardson number is a measure of the relative importance
of buoyant forces to shear forces, Ri = βgH /U 2 = N 2 /γ 2 .
The assumption of a Boussinesq fluid is that the velocity field
is divergence-free, β ≡ 0, and the Ri remains finite. No penetration
(and stress free) and fixed temperature boundary conditions
are imposed:
| y=0 = T | y=0 = 0
| y=1 = T | y=1 = 1 (10)
Because there is no dynamic boundary condition, pressure
was eliminated, resulting in the momentum (9).
3.2 Wave disturbances of permanent form
Generally, the flow and temperature fields can be arbitrarily
decomposed into background fields and disturbance fields.
To make the procedure deterministic, the assumption is made
that the disturbance field has zero mean in some sense. The
sense adopted here is that the background fields are timeaveraged
and thus steady. Further, the background fields are
prescribed and independent of the horizontal x-coordinate.
T = ˆT (y) + εT (x, y, t)
= ˆψ (y) + εψ (x, y, t) (11)
ε = A 0 /h is the dimensionless characteristic size of the disturbance
fields, where A 0 is the maximum value of the initial
disturbance. For convenience, the following representative
case of linear shear and stratification are considered:
ˆT = y
ˆψ = y2
2
(12)
Since it is the long time waveform which is required, and it
is expected that that the form of solution will be steady in a
the frame of reference of the phase velocity ν, the following
coordinate transform is made:
( ) ( )
x ξ = x − νt
→
;
t τ = εt
( ∂
∂x
=
∂ξ
∂
∂
∂t
= ε ∂
∂τ − ν ∂
∂ξ
)
(13)
It is anticipated that the approach to a wave of permanent
form occurs over a slow time scale. Substituting Eq. (11)
into Eqs. (8) and (9) yields
T ξ (y − ν) − ψ ξ = ε ( −θ τ + ψ ξ T y − ψ y T ξ
)
[
ψyξ (y − ν) − ψ ξ
]
y + RiT ξ + [ ψ ξξ (y − ν) ] ξ
= ε
(−ψ yτ + [ ] )
ψ ξ ψ yy − ψ y ψ ξy y − ψ ξξτ
( [ψξ ] )
+ ε ψ ξy − ψ y ψ ξξ
ξ
(14)
If ε ≪ 1, it is quite natural to consider the vanishing amplitude
limit of ε = 0. The basic disturbance T (0) , ψ (0) satisfies
T (0)
ξ
(y − ν) − ψ (0)
ξ
= 0
[
]
ψ (0)
(0)
yξ
(y − ν) − ψ(0)
ξ
+ RiT
y
ξ
[
+ ψ (0)
ξξ
]ξ (y − ν) = 0 (15)
The expectation that the disturbance should be wavelike in
the horizontal at any given height y is equivalent to the separation
of variables
T (0) (ξ, y) = A (ξ) θ (y)
ψ (0) (ξ, y) = A (ξ) φ (y) (16)
The system (15) then simplifies to
A ξ (θ (y − ν) − φ) = 0
A ξ
[
φy (y − ν) − φ ] y + RiθA ξ + A ξξξ (y − ν) φ = 0 (17)
The system (17) has solutions only if a separation condition
on A is satisfied, namely
A ξξξ = −k 2 A ξ (18)