Atmospheric undular bores

Abstract

We show that a recently-derived model for the propagation of nonlinear waves in the atmosphere admits undular bores as travelling-wave solutions. These solutions represent waves consisting of a damped oscillation behind a front that is preceded by a uniform breeze-type flow. The generation of such wave profiles requires a jump in the heat source across the leading front of the wave, a feature that is consistent with observations.

Keywords: 34C15; 35C07; 86A10.

Fig. 1

Photographs of some atmospheric undular bores

Fig. 2

Sketch of the oscillating behaviour of the solution $X\mapsto \widehat{f}(X)$ in Theorem 1. The figure on the left corresponds to the spiralling pattern of the solution curve $X\mapsto \mathcal{C}(X)=(\widehat{f}(X),\widehat{g}(X))$ depicted on the right in the phase-space

Fig. 3

The solution curve $X\mapsto \mathcal{C}(X)$ of the nonlinear system (4.4) through $(0,2\eta )$ spirals around the origin (the single equilibrium point of the system), with continuously diminishing distance from it as X grows towards $\infty$. The vector field points vertically upwards along the dotted branch of the vertical isocline in the half-plane $[f<0]$, vertically downward along the dashed branch in the half-plane $[f>0]$, and horizontally along the vertical axis $f=0$ (to the right on the dotted positive semi-axis and to the left on the dashed negative semi-axis). The interior $\mathcal{O}$ of the disk $[f^2+g^2=4{\eta }^2]$ is positively invariant. The connection to Fig. 2 is made by direction reversal, with $X_1$ and $X_2$ corresponding to $X_{-1}$ and $X_{-2}$ in Fig. 2

Fig. 4

Behaviour of the solution to (4.15) with initial data (4.16). The connection to Fig. 2 is made by means of the transformation (4.14) and noticing that $X_1^{†}$ and $X_2^{†}$ correspond to $X_1$ and $X_2$ in Fig. 2, respectively

Fig. 5

Sketch of an undular bore, generated when a thermal inversion occurs – with the layer of colder air (depicted in blue) overlain by a layer of warmer air (depicted in orange)

Fig. 6

Numerical solution (using Maple) of Eq. (5.9) with $f_0=0.7$, $\nu =-1$ and $X_0=0$. The wave profile propagates to the right with speed $c=\frac{10V_{-}-7V_{+}}{7}({cos}^2\alpha +\frac{{sin}^2\alpha }{C})$, where $V_{\pm }$ are the breeze speeds