= Solution
For this sign of the <viscous Burgers equation>, the <Cole-Hopf transformation> is $q=2\delta\partial_\theta\log\psi$. Integrating the initial sinusoid gives $\psi(\theta,0)=C\exp[-\cos\theta/(2\delta)]$; the positive multiplicative constant cancels from $q$. Set $a=1/(2\delta)$ and $s=\delta Z$. The <Fourier series> of the initial exponential and the <heat equation> yield the exact <sinusoidal Cole-Hopf solution for negative-flux Burgers flow>:
$$
\psi(\theta,Z)=I_0(a)+2\sum_{n=1}^{\infty}(-1)^nI_n(a)e^{-n^2s}\cos n\theta.
$$
Each harmonic decays by its heat multiplier, which can also be obtained directly by convolving it with the Gaussian <heat-kernel convolution>. At $s\gg1$ the first harmonic dominates:
$$
\psi=I_0(a)-2I_1(a)e^{-s}\cos\theta+O(I_0(a)e^{-4s}),\qquad
\psi_\theta=2I_1(a)e^{-s}\sin\theta+O(I_0(a)e^{-4s}).
$$
The error bounds use $|I_n(a)|\le I_0(a)$ for real positive $a$, also immediate from the Fourier-integral representation. Dividing, including the correction from the denominator, gives
$$
q=4\delta\frac{I_1(a)}{I_0(a)}e^{-s}\sin\theta+O(\delta e^{-2s}).
$$
For $\delta\ll1$, the <large-argument asymptotic expansion of a modified Bessel function> gives $I_1(a)/I_0(a)\to1$. Hence
$$
\boxed{q(\theta,Z)\sim4\delta\sin\theta\,e^{-\delta Z}.}
$$
The sign is positive because the first Fourier harmonic of $\psi$ is negative, so its angular derivative is positive. For a fixed nonzero $\sin\theta$, the relative corrections vanish as $\delta\to0$ and $s\to\infty$; at the symmetry points both the exact solution and the leading expression vanish.
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