Static solutions

A notebook to plot the analytic 'static' solutions (or at least late time solutions) found in Spitkovsky et al. 2002 and compare them against static solutions given by numerical code.

Localised heating in a rotating atmosphere without friction

In Spitkovsky et al. 2002, it's given that for a constant delta function heating $Q(x,y) = Q_0\delta(x)$, the fluid follows a growing vortex sheet solution: $$ h(x,t) = \frac{Q_0 t}{2 a_R} e^{-|x|/a_R}, $$ $$ v_x(x,t) = \text{sign}(x) \frac{Q_0}{2H_0}e^{-|x|/a_R}, $$ $$ v_y(x,t) = - \text{sign}(x) \frac{Q_0 t f}{2H_0}e^{-|x|/a_R}, $$ where $H_0$ is the scale height, $a_R$ the Rossby adjustment radius and $f = 2\Omega\cos\theta$ the local Coriolis parameter at latitude $\theta$ for a body with angular frequency $\Omega$.



In [18]:

    
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation
from IPython.display import HTML
%matplotlib inline



In [60]:

    
Q0 = 0.2
aR = 1.0
f = 0.2
H0 = 1.0

nx = 100
nt = 20

xs = np.linspace(-5.0, 5.0, nx, endpoint=True)
ts = np.linspace(0.0, 10, nt+1, endpoint=True)
qs = np.zeros((nt+1, nx, 3))



In [61]:

    
for i in range(nt+1):
    qs[i,:,0] = (Q0 * ts[i] / (2.0 * aR)) * np.exp(-np.abs(xs) / aR)
    qs[i,:,1] = np.sign(xs) * (Q0 / (2.0 * H0)) * np.exp(-np.abs(xs) / aR)
    qs[i,:,2] = - np.sign(xs) * (Q0 * ts[i] * f / (2.0 * H0)) * np.exp(-np.abs(xs) / aR)



In [68]:

    
fig = plt.figure(figsize=(8,6))
ax = fig.gca()
plt.rc("font", size=15)

def animate(i):
    ax.clear()
    ax.plot(xs, qs[i,:,0], xs, qs[i,:,1], '--', xs, qs[i,:,2], '-.', lw=2)
    title = 't = ' + str(ts[i])
    ax.set_title(title)
    ax.set_xlabel(r'$x$')
    ax.legend([r'$h$', r'$v_x$', r'$v_y$'])#, loc='center left', bbox_to_anchor=(1, 0.5))
    ax
    ax.set_ylim([-0.2, 0.8])    

anim = animation.FuncAnimation(fig, animate, frames=nt+1, interval=150)



In [69]:

    
HTML(anim.to_html5_video())









    Out[69]:



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