T-II-1

EDP: Waves breaking, Burgers and Shocks

Far from the beach , i.e., when the sea is deep, water waves travel on the surface of the sea and their shape is not deformed too much. When the waves approach the beach their shape begins to change and eventually breaks upon touching the seashore.

Here we are interested in studying a simple model that captures the essential features of this physical system. More precisely we will examine the Burgers equation, a non-linear equation introduced as a toy model for turbulence. This equation can be explicitly integrated and, in the limit of zero viscosity, its solution generally displays discontinuities called shocks.

Model 0 : Deep sea

Consider the following equation for the shape of the wave:

$\partial _{t}u(x,t)=c\partial _{x}u(,t)$

where $u(x,t)$ is the height of the wave with respect to the unperturbed sea level, and c is constant. We will use the following initial condition $u(x,t=0)=u_{\text{init}}(x)$

$u_{\text{init}}(x){\begin{cases}0\quad {\text{for}}\;|x|>1\\1-x^{2}\quad {\text{for}}\;|x|<1\end{cases}}\quad \quad \quad (0)$

• Find the solution of the latter equation and compute the velocity of the wave
• Discuss why this model may mimic wave motion in deep sea.

Model 1 : Approaching the sea shore

It is possible to show that, when the height of the wave is negligible as compared to the sea depth, the velocity of the wave does not depend on $u$. On the contrary, when approaching the sea shore this is not anymore true and the velocity becomes proportional to $u$:

$v(u)\simeq c_{1}u+v_{0}$

where $c_{1}$ is a constant and $v_{0}$ is the velocity of the bottom of the wave. Let us take $c_{1}=1$ and write the equation for the wave motion in the frame of the bottom of the wave, which yields:

$\partial _{t}u(x,t)=u(x,t)\partial _{x}u(x,t).\quad \quad \quad (1)$

This is the non linear equation originally introduced by Burgers. In Burgers' derivation a viscous term was also present

$\partial _{t}u(x,t)=u(x,t)\partial _{x}u(x,t)+\nu \partial _{x}^{2}u(x,t),\quad \quad \quad (2)$

$\nu$ being the viscosity, and Eq. (1) is called the inviscid limit of Eq. 2.

Our strategy will be to investigate the properties of Eq1 by resorting to the method of characteristics. Moreover, we will integrate Eq 2 by using the Cole Hopf transformation, and then recover Eq.1 by taking the limit $\nu \to 0$.

The Method of characteristics

• Write the solution of Eq.1 with the initial condition of Eq.0.
• Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,t)=0} is alway a solution for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | x | > 1 } . Denote by $u_{0}(x,t)$ this solution.
• Show that two other solutions exist, namely:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\pm}(x,t)= \frac{1}{4 t^2} \left( 1 \pm \sqrt{\Delta(x,t)}\right)^2-1 }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta(x,t)= 4 t^2 -4 x t +1 } .

• Show that for $t<0.5$ only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_0(x,t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_-(x,t)} are physically acceptable.
• Draw the full solution for $t=0.5$ and compute its derivative at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=1 }
• Show that for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t>0.5 } the solution is a multivalued function $u_{\text{mult}}(x,t)$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\text{mult}}(x,t)= \begin{cases} u_0(x,t) \quad \text{ for} \; | x | > 1 \\ u_-(x,t) \quad \text{ for} \; -1< x < \frac{1}{4 t}+t \\ u_+(x,t) \quad \text{ for} \; 1< x < \frac{1}{4 t}+t \end{cases} }

and draw some examples of solutions for various values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .

We look now for a solution which is a single-valued function. For $t>0.5$ this function should display a discontinuity called shock. We will show in the second part of this exercise that the solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_+(x,t)} is always unstable. The single valued solution for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t>0.5 } takes the form

$u_{\text{single}}(x,t)={\begin{cases}u_{0}(x,t)\quad {\text{ for}}\;x<-1\\u_{-}(x,t)\quad {\text{ for}}\;-1<x<x_{\text{shock}}(t)\\u_{0}(x,t)\quad {\text{ for}}\;x>x_{\text{shock}}(t)\end{cases}}$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\text{shock}}(t) } indicates the shock location. The position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\text{shock}}(t) } is imposed by a conservation law that should be valid at all times.

• Find the conservation law for the wave and determine Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\text{shock}}(t) } .

The Cole Hopf Transformation

Consider Now Eq.2 with $\nu >0$. This equation can be integrated via the Cole Hopf transformation which amounts to perform the following change of variable:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = -2\nu \frac{\partial}{\partial x} \ln\phi }

• Derive, from the Burgers equation in Eq. (3), the equation satisfied by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi } . This can be done in three steps :

(i) $v=-2\epsilon \partial _{x}\psi$, (ii) integrate the equation that you have obtained, (iii) substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi = \ln \phi } .

• Show that, in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu \to 0 } , the profile $u(x,t)$ can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(x,t)= \frac{x-y_{\text{min}}(x)}{t} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{\text{min}}(x) } is the location of the global minimum of the function

$E_{x,t}(y)={\frac {1}{2t}}\,(x-y)^{2}+V(y)$

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_y V(y) = u_{\text{init}}(y) } . The initial condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\text{init}}(y) } is given in Eq.0.

• Show that the solutions $u_{0}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_+ } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_- } can be interpreted as the points where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{x,t}(y) } is stationary, i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_y E_{x,t}(y)= 0 }

• Draw Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{x,t}(y) } at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=1} for different value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1< -1, \quad -1< x_2<1, \quad 1<x_3<x_s(1), \quad x_4=x_s(1), \quad x_s(1)< x_5<5/4, \quad x_6>5/4 }

• Comment on the obtained results.