48
J. Grue
 (W
(1)
I
) = −kT 0  (𝜙 0I ) −
 (W F )
C 0
,
(7)
 (W
(2)
I
) = kT 0  (𝜂W
(1)
I
) − i𝐤 ⋅  (𝜂∇ H 𝜙 0I ) +
i𝐤 ⋅  (𝛽∇ H 𝜙 0F )
C 0
,
(8)
 (V
(1)
I
) = kT 1  (𝜙 1I ),
(9)
 (V
(2)
I
) = −kT 1 (𝜂V
(1)
I
) − i𝐤 ⋅  (𝜂∇ H 𝜙 1I ),
(10)
 (𝜙
(1)
0F
) =
 (𝜙 0I )
C 0
−
T 0  (W F )
k
,
(11)
 (𝜙
(2)
0F
) = −
 (𝜂W
(1)
I
)
C 0
+
T 0 i𝐤
k
⋅  (𝛽∇ H 𝜙 0F ) −  (𝛽W F ).
(12)
Here,  denotes Fourier transform,  −1 inverse transform, 𝐤 = (k 1 , k 2 ) wavenumber
vector in Fourier space and k = |𝐤|. Further, T 0 = tanh(kh 0 ), T 1 = tanh(kh 1 ) and
C 0 = cosh(kh 0 ). In (8) and (12) 𝜙 0F = 𝜙
(1)
0F
+ 𝜙
(2)
0F
.
Time-Integration
The interfacial motion is expressed by the difference and sum potentials along the
interface:
𝛹 (𝐱, t) = 𝜙 1I (𝐱, t) − 𝜇𝜙 0I (𝐱, t), Φ(𝐱, t) = 𝜙 1I (𝐱, t) + 𝜙 0I (𝐱, t), at I,
(13)
where 𝜇 = 𝜌 0 ∕𝜌 1 denotes the density ratio. Note that the jump condition in the
former of the equations in (13) accounts for interactions between the density jump
and ambient pressure fields. The interfacial elevation and jump in potential along I
are integrated forward in time using the kinematic and dynamic boundary conditions
at the interface, giving
𝜂 t = V I = W I , 𝛹 t + g
′
𝜂 =   2 , at I.
(14)
The r.h.s. of the latter equation in (14) is:
  2 = −
|∇ H 𝜙 1I |
2
− 𝜇|∇ H 𝜙 0I |
2
− (1 − 𝜇)W
2
I
2 + 2|∇ H 𝜂| 2
−
−2W I ∇ H 𝜂 ⋅ ∇ H (𝜙 1I − 𝜇𝜙 0I ) + |∇ H 𝜂 × ∇ H 𝜙 1I |
2
− 𝜇|∇ H 𝜂 × ∇ H 𝜙 0I |
2
2 + 2|∇ H 𝜂| 2
, (15)
and is evaluated in the nonlinear calculations. In the linear calculations,   2 is put
to zero. An RK4-scheme is used for the time integration of the Fourier transformed
versions of (14).