How can vorticity tilt? Most descriptions of tornado genesis make a reference to vorticity at the ground level with a horizontal axis somehow tilting to the vertical. Why doesn't this violate conservation of angular momentum?
 A: A vortex in a uniform flow field will not tilt, it will stay as it is aligned and move along with the flow. However, if there is a gradient in the velocity field, this can cause a vortex to stretch and/or tilt.
To get math-y first... if we take the equation for the velocity of a fluid and we take the curl of that equation, we get what is called the vorticity equation. This equation is an expression of the conservation of (local) angular momentum. If we assume the fluid is incompressible (which just makes the equation simpler to look at by removing some terms, but it doesn't change what we're talking about here), we get:
$$ \frac{D \vec{\omega}}{Dt} = (\vec{\omega} \cdot \nabla) \vec{u} + \nu \nabla^2 \vec{\omega} $$
where $\vec{\omega}$ is the vorticity. The second term on the right hand side is the viscous dissipation of vorticity. The first term on the right hand side is the stretching/tilting of a vortex due to velocity gradients.
Let's look at what would make a horizontal vortex tilt to a vertical one like a tornado would. Let's assume that our ground plane is in the X-Y direction and our altitude (vertical) direction is in Z. A horizontal vortex that is up in the clouds would be the X-component of vorticity, $\omega_x$. A vertical vortex like a tornado would be the Z-component of vorticity, $\omega_z$.
Let's write out those two specific equations to see how they might interact. We are only interested in the tilting, so let's ignore viscosity for the moment. The horizontal vortex equation looks like:
$$ \frac{D \omega_x}{Dt} = \omega_x \frac{\partial u_x}{\partial x} + \omega_y \frac{\partial u_x}{\partial y} + \omega_z \frac{\partial u_x}{\partial z} $$
and the vertical equation looks like:
$$ \frac{D \omega_z}{Dt} = \omega_x \frac{\partial u_z}{\partial x} + \omega_y \frac{\partial u_z}{\partial y} + \omega_z \frac{\partial u_z}{\partial z} $$
Now, let's assume that we start out with no vertical vorticity -- no tornado like behavior, just a big horizontal one. This means the initial $\omega_y$ and $\omega_z$ are zero. If we simplify the equations, we get:
$$ \frac{D \omega_x}{Dt} = \omega_x \frac{\partial u_x}{\partial x} $$
$$ \frac{D \omega_z}{Dt} = \omega_x \frac{\partial u_z}{\partial x} $$
Okay -- so what does this tell us. Well, first it tells us that the horizontal vorticity (initially) will increase or decrease as it gets stretched or squeezed along the axis of the vortex. If you stretch it, the vortex gets smaller in diameter and vorticity increases -- this is the "spinning figure skater pulling in their arms and speeding up" effect. The opposite happens if you squish it -- it gets wider and spins slower.
The second equation tells us that the vertical vortex will start to increase due to gradients in the vertical velocity along the horizontal direction. In weather, this means there is an updraft of down-draft. In other words, you have horizontal blowing wind that turns upwards and creates gradients in the vertical velocity along the axis of the tornado. This will cause the vortex to start tilting, and turns the horizontal vorticity into vertical vorticity.
During this tilting process, vorticity (and angular momentum) is still conserved, it is just re-arranged between components. So the vorticity in the horizontal direction will decrease, vorticity in the vertical will increase, and it is all driven by gradients in the velocity aligned with the vortex itself.

Just to complete the picture, once the vortex starts to tilt and we have both X and Z components of vorticity, the equations becomes:
$$ \frac{D \omega_x}{Dt} = \omega_x \frac{\partial u_x}{\partial x} + \omega_z \frac{\partial u_x}{\partial z} $$
$$ \frac{D \omega_z}{Dt} = \omega_x \frac{\partial u_z}{\partial x} + \omega_z \frac{\partial u_z}{\partial z} $$
and this shows the interchange between vorticity components based on the velocity field. It is obviously a dynamic system so it isn't obvious how it all plays out, but an updraft will cause the horizontal vortex to start to tilt towards the vertical, making a tornado start to form. Then, once you have some vertical vorticity $\omega_z \neq 0$, that vortex can get stronger or weaker due to stretching (faster spin, smaller diameter) or squeezing (slower spin, larger diameter) caused by $\frac{\partial u_z}{\partial z}$ -- i.e. due to acceleration or deceleration of the updraft along the axis of the tornado, respectively.
A: A tornado is not an isolated system and may exchange angular momentum with the Earth and with the non-tornado parts of Earth’s atmosphere.
