Unnormalized multivariate Gaussian?

#1
I'm looking at bivariate Gaussian variables centered on zero.

The chi distribution provides moments for a normalized multivariate Gaussian -- i.e., a random variable \(Y = \sqrt{\sum (\frac{X_i}{\sigma_i}})^2\).

But I want the variance for the unnormalized vectors \(Y' = \sqrt{\sum X_i^2}\), so I can see the relationship with the constituent variances.

Is there a canonical distribution for that? If not how can we derive that unnormalized variance?