Weakly nonlinear waves in stratified shear flows (Q2147894)

The authors consider the two-dimensional Euler equations for incompressible and divergence-free flows in a layered domain written after a change of variables, a rescaling and in non-dimensional form as: \(u_{t}+(U+\varepsilon u)u_{x}+w(U^{\prime }+\varepsilon u_{z})=-\frac{1}{1+r}p_{x}\), \( w_{t}+(U+\varepsilon u)w_{x}+\varepsilon ww_{z}=-\frac{1}{1+r}p_{z}\), in \( -d<z<-1+\varepsilon H(x,t)\), \(u_{x}+w_{z}=0\), in \(-d<z<\varepsilon \eta (x,t) \), \(P+\varepsilon p=P_{0}\), \(w=\eta _{t}+(U+\varepsilon u)\eta _{x}\), on \( z=\varepsilon \eta (x,t)\), \(P_{+}+\varepsilon p_{+}=P_{-}+\varepsilon p_{-}\) , \(w_{\pm }=H_{t}+(U+\varepsilon u_{\pm })H_{x}\), on \(z=-1+\varepsilon H(x,t) \), \(w=0\) on \(z=-d\), where \(U\) is a background current. They introduce a linearization of this problem and letting \(\varepsilon \rightarrow 0\) they obtain: \((U-c)u_{x}+wU^{\prime }=-p_{x}\), \(0=p_{z}\), in \(-1<z<0\), \( (U-c)u_{x}+wU^{\prime }=-\frac{p_{x}}{1+r}\), \(0=p_{z}\), in \(-d<z<-1\), \( u_{x}+w_{z}=0\), in \(-d<z<0\), \(p=\eta \), \(w=(U-c)\eta _{x}\), on \(z=0\), \( p_{+}-p_{-}=-rH\), \(w=(U-c)H_{x}\), on \(z=-1\), \(w=0\) on \(z=-d\). They look for long wave solutions to this linearized problem in terms of arbitrary functions \(\eta (x)\) and \(A(x)\) for \(p\) and \(w\) in \(z\in \lbrack -1,0]\). They obtain the Burns condition from which they derive an integral form for the dispersion relations. They consider special cases where these dispersion relations are explicit. Now looking for weakly nonlinear long waves, the authors introduce the change of variables \(\xi =\sqrt{\varepsilon }(x-ct)\), \(\tau =\varepsilon ^{3/2}t\); and the scaling \(w\mapsto \sqrt{ \varepsilon }w\). They formally expand the variables \(u,w,p,H\) near \( \varepsilon =0\) and they derive the zero- and first-order systems that they solve. This allows to derive Korteweg-de Vries models for the free surface and for the velocity field and pressure and the authors give few examples.