2. Formulation of the Problem
We consider a shear flow that consists of an upper layer fluid flowing with a uniform velocity over a lower layer fluid of higher density as shown schematically in
Figure 1. We assume that both of the layers are of infinite extents in the two-dimensional Cartesian coordinate system. The classical examples are air-flow over a free surface of the water or a motion of one layer of a fluid relative to another layer of a different density. In the present study, the effect of surface tension is taken into consideration at the interface between the fluid layers by assuming that the two fluids are immiscible, and the fluids of both layers are perfect. We choose coordinate axes as shown in
Figure 1, with the
x-axis being directed along the horizontal direction and the
z-axis being directed vertically upward, so that the acceleration due to gravity
is directed downward, with the line
being along the mean interface.
Let be the density of the upper layer fluid, be the density of the lower layer fluid, U be the fluid speed of the upper layer (wind speed), and T be the surface tension at the interface of the two-layers.
The assumption of perfect fluids allows us to introduce hydrodynamic potentials for the perturbed velocities in each layer as
, for
, where subscript 1 corresponds to the upper layer fluid and the subscript 2 corresponds to the lower layer fluid. The condition of fluid incompressibility yields the Laplace equation for the potential flow in each layer, which is given by the equation:
We assume that there are no wave perturbations far from the interface in the vertical direction as
:
The kinematic boundary conditions at the interface are given by the set of equations
where
is the perturbation of the interface.
The dynamic boundary condition at the interface
yields:
In the linear approximation of wave perturbations of infinitesimal amplitudes, we can seek a solution for the perturbation at the interface in the form:
where
stands for the complex-conjugate, and
A is a constant.
The perturbations of the velocity potentials in the upper and lower layers, which satisfy the Laplace equations and kinematic boundary conditions as in Equations (
4) and (5) in the linear approximation, are given by:
and
The boundary conditions of vanishing perturbations at
are taken into account in this approximation. Substituting these solutions into the dynamic boundary condition (
6), we obtain the dispersion relation:
Solving the dispersion relation (
10) with respect to
, we obtain the dependences of the wave frequency
on the wavenumber
k in the explicit form (as in [
2]):
where
is the density ratio.
Graphical representations of the dispersion relations for three typical velocities are shown in
Figure 2. For the sake of convenience, we assume that wavenumber
k is positive and that the frequency
is of either sign. However, from the physical point of view, the wave frequency
is a positive quantity, whereas the wavenumber
k is of either sign being in the interval
.
Equation (
11) describes the two branches of the dispersion relation that are symmetric with respect to the
k-axis for
. These branches correspond to capillary-gravity waves travelling in opposite directions with phase speed
(
Figure 2a).
For
, the dispersion curves become non-symmetric because of the wave drift caused by the flow. When
U increases and becomes bigger than some critical value
, the lower branch of the dispersion curve in
Figure 2b changes sign and becomes positive for
where
Here, the negative sign pertains to
and the positive sign to
. The portion of the dispersion curve in
Figure 2 where the frequency changes sign corresponds to NEWs [
2]. The dispersion curves shown in
Figure 2 are presented in
Figure 3 for
with
. In this representation, the wave energy becomes negative when the wave frequency
becomes formally negative. However, the waves with ‘negative energy’ are actually captured by a fluid flow and propagate co-current as shown in
Figure 3.
When
U increases further, the two branches of the dispersion curve continue to converge, and eventually reconnect (lines 4 in
Figure 2b) when
, where
, for an air-water interface with
,
. The KH instability arises when
U becomes greater than
, and the instability occurs in the interval
, where
Such re-connection is typical for the interaction of waves of opposite energy signs [
2,
3]. Thus, the appearance of the KH instability can be attributed to an interaction of waves of opposite energy signs. The NEWs on the lower branch of the dispersion curve transfer their energy to the positive-energy waves, on the upper branch of the dispersion curve. As a result, the amplitudes of both waves grow with time.
When
U is in the interval,
, then there is no KH instability; however, there are non-growing but potentially unstable NEWs. Negative energy waves existing in the interval
can grow if the associated wave energy is depleted, i.e., if there is a mechanism that draws their energy. It can be easily shown from the expressions for
and
that these velocities are very close to each other when
a is small. This is the case of the air-water interface with
. If
, which is typical for internal layers in the ocean or atmosphere, then
which was initially noted by Benjamin [
8]. There are potentially different mechanisms which can cause NEWs to grow. In particular, viscous dissipation in an immovable lower fluid layer leads to the growth of NEWs [
12]. This is analogous to dissipative instability in plasma physics [
13].
For NEWs to be amplified, the viscosity must lead to ‘positive losses’. For example, NEWs in the model under consideration will be damped if the moving upper layer is viscous rather than the fixed lower layer. However, positive-energy waves can grow on the upper branch of the dispersion curve, since the viscosity of the moving upper layer leads to ‘negative damping’.
Indeed, in passing over to the reference system in which the upper layer is at rest and the lower layer is moving with a uniform speed
U, the energy of the growing mode and, simultaneously, the dissipation change their signs [
2]. In such a reference system, NEWs exist in the upper branch of the dispersion curve and can grow under the influence of positive dissipation. The shear flow instability associated with this mode does not vary with the reference system. The dispersion relation in the reference frame in which the upper layer is at rest and the lower layer moves with the speed
U in the opposite direction can be obtained easily from the dispersion relation (
10) by the formal replacement
. In this case, NEWs arise when the velocity of the lower layer
, where
It may be noted that the difference between the critical velocities and is small if (as in the case of internal waves on the ocean pycnocline), but it is rather significant for (as in the case of air-water interface in which m/s). In general, NEWs can exist on both the branches of the dispersion curves for waves that are slowed down relative to the flow which means their phase velocities being less than the velocities of the corresponding fluid layers.
The dispersion dependence (Equation (
11)) can be considered in the ‘mean-mass’ reference frame which is moving with the velocity
. In this frame, the lighter fluid in the upper layer moves in the positive direction with velocity the
, whilst the heavier fluid in the lower layer moves in the opposite direction with the velocity
. In this case, the two branches of the dispersion curves are symmetric relative to the
k-axis due to the Doppler frequency shift
(
Figure 4). It can be noted that there are no NEWs in this reference frame and the dispersion curves do not change their signs.
For sufficiently large values of
U, there are portions on the dispersion curves that correspond to the ‘retarded waves’. The phase speed of these waves is less than the current speed of the corresponding layer. For
, the retarded waves appear on the dispersion curve below the dashed straight line
(
Figure 4). Waves with the wavenumbers in the range
, where
have phase speeds
in the “mean-mass” reference frame.
With further increase of the current speed, the retarded waves appear for
on the lower branch of the dispersion curve; they are shown in
Figure 4b above the straight line
in a very narrow wavenumber range
with
being given in Equation (
12). It is difficult to demonstrate both the dashed straight lines having different slopes and the portions of dispersion curves corresponding to the retarded waves in the figure as the parameter
a is considered to be very small for the air-water interface. For this reason, they are shown in separate frames in
Figure 4 with different horizontal and vertical scales.
The estimates for the air-water interface demonstrate that
m/s, whereas
m/s which is equal to the minimal phase speed of surface waves on quiescent water. For
, the air viscosity provides ‘negative dissipation’, which may give rise to the dissipative instability of surface waves [
2,
3,
13]. However, the dissipation in water continues to be positive up to the velocity
. Thus, the dissipation in water increases the threshold velocity of the wind and, consequently, delays the onset of instability of wind waves [
7].
Furthermore, with the increase in the wind velocity, the effective dissipation in the lower layer fluid changes its sign for
and then for
with
the dispersion branches reconnect, and the KH instability arises (see lines 3, 4 and 5 in
Figure 2b). In real conditions, wind wave instability sets in much earlier than the classical KH theory predicts [
7], for
m/s.
From the aforementioned discussion, it reveals that, for , the heavier lower layer fluid ‘carries’ those surface waves of which the dispersion properties (for ) are perturbed slightly due to the presence of the lighter fluid in the upper layer. Simultaneously, the dissipation in the air which moves faster than certain waves at the interface, leads to instability of surface waves, i.e., to the appearance of wind waves on quiescent water.