The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, Because the fluid is assumed incompressible, this velocity field has the streamfunction representation
where the subscripts indicate partial derivatives. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence . In the streamfunction representation, Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz
where is a spatial wavenumber. Thus, the problem reduces to solving the equation
The domain of the problem is the following: the fluid with label 'L' lives in the region , while the fluid with the label 'G' lives in the upper half-plane . To specify the solution fully, it is necessary to fix conditions at the boundaries and interface. This determines the wave speed c, which in turn determines the stability properties of the system.
The first of these conditions is provided by details at the boundary. The perturbation velocities should satisfy a no-flux condition, so that fluid does not leak out at the boundaries Thus, on , and on . In terms of the streamfunction, this is
The other three conditions are provided by details at the interface .
Continuity of vertical velocity: At , the vertical velocities match, . Using the stream function representation, this gives
Expanding about gives
where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.
The free-surface condition: At the free surface , the kinematic condition holds:
Linearizing, this is simply
where the velocity is linearized on to the surface . Using the normal-mode and streamfunction representations, this condition is , the second interfacial condition.
Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at is given by the Young–Laplace equation:
where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is
Thus,
However, this condition refers to the total pressure (base+perturbed), thus
(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form
this becomes
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations,
with to yield
Putting this last equation and the jump condition on together,
Substituting the second interfacial condition and using the normal-mode representation, this relation becomes
where there is no need to label (only its derivatives) because
at
- Solution
Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation with the boundary conditions has the solution
The first interfacial condition states that at , which forces The third interfacial condition states that
Plugging the solution into this equation gives the relation
The A cancels from both sides and we are left with
To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then,
and clearly
- If , and c is real. This happens when the lighter fluid sits on top;
- If , and c is purely imaginary. This happens when the heavier fluid sits on top.
Now, when the heavier fluid sits on top, , and
where is the Atwood number. By taking the positive solution, we see that the solution has the form
and this is associated to the interface position η by: Now define