Modulation instability and rogue waves for shear flows with a free surface

We study free surface gravity waves in the presence of a depth-dependent shear current with a nonzero vorticity gradient. The evolution of weakly nonlinear, narrow-band wave packets is governed by a nonlinear Schrödinger equation. When dispersion and nonlinearity are of the same (opposite) signs, modulation instability will be present (absent), and rogue waves represented by breathers can (cannot) occur, respectively. For irrotational flows, rogue waves only occur for sufficiently deep water, or more precisely, $kh>1.363$, where $k$ is the wave number of the carrier envelope and $h$ is the water depth. While the irrotational and linear shear current cases have been treated previously, the present study demonstrates the importance of a shear current with a nonzero vorticity gradient. For a concave current, the threshold for modulation instability of a wave packet moving with (or against) the shear current will reduce (or increase) the numerical bound of 1.363, respectively. The opposite will hold for the case of a convex current. The growth rate of a disturbance will also be larger for concave currents in comparison with convex and linear currents. The streamline patterns for the transient case of rogue waves will be illustrated for the simple case of a linear shear current. Shear currents near the sea surface are known to have a profound influence on the dynamics of wind-generated waves, and the present investigation bears on this.

Figure 1

A schematic view of a wave packet propagating in a shear flow with nonzero vorticity gradient. The upper surface ($y=1$) is free, but the bottom ($y=0$) is flat and rigid.

Figure 2

Relation between the input setting [carrier wave number ($k$), water depth ($h$)] and the existence condition of rogue waves ($\beta \gamma >0$) for the backward mode of a “weak” linear current: (i) the current $U\left(y\right)=0.1y$; (ii) the product $\beta \gamma$ versus $kh$; (iii) the group velocity ($c_g$) versus $kh$; (iv) the phase speed ($c$) versus $kh$. Both $c$ and $c_g$ need to be outside the range of the shear velocity ([0, 0.1] here) to avoid critical level singularities.

Figure 3

Relation between the input setting [carrier wave number ($k$), water depth ($h$)] and the existence condition of rogue waves (βγ>0) for the backward mode of an “intermediate” linear current: (i) the current $U\left(y\right)=0.5y$; (ii) the product βγ versus $kh$; (iii) the group velocity ($c_g$) versus $kh$; (iv) the phase speed ($c$) versus $kh$. Both $c$ and $c_g$ need to be outside the range of the shear velocity ([0,0.5] here) to avoid critical level singularities.

Figure 4

The dynamics of modulation instability for a forward mode of a concave shear current: (i) the background velocity profile; (ii) product of the dispersive (β) and nonlinear (γ) coefficients, positive for rogue wave occurrence; (iii) variation of the group velocity ($c_g$) with $kh$; (iv) variation of the phase velocity ($c$) with $kh$. Both $c$ and $c_g$ need to be outside the range of the shear velocity ([0,0.5] here) to avoid critical level singularities.

Figure 5

The dynamics of modulation instability for a backward mode of a concave shear current: (i) the background velocity profile; (ii) product of the dispersive (β) and nonlinear (γ) coefficients, positive for rogue wave occurrence; (iii) variation of the group velocity ($c_g$) with $kh$; (iv) variation of the phase velocity ($c$) with $kh$. Both $c$ and $c_g$ need to be outside the range of the shear velocity ([0,0.5] here) to avoid critical level singularities.

Figure 6

The dynamics of modulation instability for a forward mode of a convex shear current: (i) the background velocity profile; (ii) product of the dispersive (β) and nonlinear (γ) coefficients, positive for rogue wave occurrence; (iii) variation of the group velocity ($c_g$) with $kh$; (iv) variation of the phase velocity ($c$) with $kh$. Both $c$ and $c_g$ need to be outside the range of the shear velocity ([0,0.5] here) to avoid critical level singularities.

Figure 7

The dynamics of modulation instability for a backward mode of a convex shear current: (i) the background velocity profile; (ii) product of the dispersive (β) and nonlinear (γ) coefficients, positive for rogue wave occurrence; (iii) variation of the group velocity ($c_g$) with $kh$; (iv) variation of the phase velocity ($c$) with $kh$. Both $c$ and $c_g$ must be outside the range of the shear velocity to avoid critical level singularities.

Figure 8

The variation of the maximum growth rate with changing strength of a linear current $U\left(y\right)=by$ for $kh=1.5$. This growth rate (γ) is normalized by the corresponding growth rate in the absence of shear (${\gamma }_{\mathrm{no}\mathrm{shear}}$).

Figure 9

The variation of the maximum growth rate with changing strength of concave currents $U\left(y\right)=b{y}^2$ for varying $b$ and $kh=1.5$. This growth rate (γ) is normalized by the corresponding growth rate in the absence of shear (${\gamma }_{\mathrm{no}\mathrm{shear}}$).

Figure 10

The variation of the maximum growth rate with changing strength of convex currents $U\left(y\right)=b(y-0.5y^2)$ for varying $b$ and $kh=1.5$. This growth rate (γ) is normalized by the corresponding growth rate in the absence of shear (${\gamma }_{\mathrm{no}\mathrm{shear}}$).

Figure 11

Streamline patterns for a rogue wave for a backward mode in a linear shear flow $U\left(y\right)=0.4y$ (mode going against the current). (a) Time $t=\pm 10000$, (b) $t=0$, with input parameters $S_0=1, \varepsilon =0.03, k=2, h=1, c=-0.397, c_g=-0.0104, \beta =0.0948, \gamma =19.067$.