How can it be shown that turbulence occurs in fluid dynamics?
I think people imply that it develops because of the $\text{rot}$ terms in the equations of motion, i.e. the Navier-Stokes equations, but I don't see how. Of course, these cross-derivatives are large for a curled up vector field, but why do these expressions force the fluid on an inward spiraly trajectory, and then even multiples of these?
Is there maybe an illustrative explicit calculation in two dimensions?
There is a simple general argument for why you get small-scale motion from large-scale motion in any nonlinear nonintegrable continuous mechanical system, whether it is fluids, or electromagnetic waves interacting with charged plasmas, or surface waves on water, or anything nonlinear at all. This argument must break down for those special cases where the usual turbulence doesn't occur, like 2D fluids.
The reason is the ultraviolet catastrophe--- the idea that to get to thermal equilibrium, all modes have to have the same amount of energy. Any mechanical system is only in statistical equilibrium when all its modes have about the same amount of energy. This Boltzmann equilibrium is therefore unattainable for smooth motions of continuous fields, because it requires that you divide a finite energy between infinitely many modes, most of which involve very short wavelengths.
The finite-energy statistical equilibrium for any continuous field is then a zero temperature state where all modes contain an infinitesimal amount of energy, which is the partition of the initial energy. The energy cascade is the method by which a fluid tries to do the paritition, by sending energy down into short wavelength modes in a random looking way, to get closer to the statistical equilibrium state. Since this is impossible, you just get a continuous draining of energy from long-wavelength motion to short wavelength motion, and when this draining process reaches a scale-invariant steady state, we call the situation isotropic homogenous turbulence.
There is always damping in a physical system, and damping drains energy into molecular motion in one step, not by a nonlinear cascade, just by thermodynamically converting the energy to heat. This one-step process is only relevant in fluids at short wavelengths, because it goes like the gradient of the velocity. A uniform velocity carries energy, but by Galilean invariance, it has no dissipative damping.
Because of Galilean invariance, in fluids there is an arbitrarily large separation of scales between the distance scales where nonlinearity is important and the much smaller scales where damping is important. Inbetween, you get a regular nonlinear mixing which drains energy from long wavelength modes to short wavelength modes, without significant damping, and in a statistically random way, because any tiny perturbation to the long-wavelength modes will produces a completely different short wavelength modes, because the process is spreading out into the much large phase space of the short wavelength modes, in an unstable, chaotic, way.
This argument only fails in certain special cases. It fails generically in 1+1 dimensions, when space is a line, because in a line there are only two modes at any wavenumber k. You still have an inaccessible Boltzmann state, because there are infinitely many k, but there is no growth in the number of modes with k, as there is in 2D and higher. So energy is just as likely to move to smaller k as to larger k, and if you have turbulence, it is more like energy diffusion, where the energy random walks from smaller to larger k, without any particular reason to get to larger k except if it randomly happens to get there.
This means that you can easily have energy in a certain number of k modes bound up in a closed motion, and this is reflected in the fact that thermalization in homogenous 1D systems with local nonlinear interactions is difficult. You run into a lot of soliton solutions, and other special states, where the energy just refuses to get random, but is nonlinearly shared in a non-thermal way between a bunch of low-k modes. This was discovered by Fermi Pasta and Ulam, when they tried to simulate the approach to statistical equilibrium in a 1d system using an early computer. Instead of thermalization, they discovered that their model never reached equilibrium, and this was a major motivation for the study of one dimensional integrable systems.
But 1d systems, as interesting as they are, are rare. This sort of nonsense doesn't happen often in 2d and above, because there are just so many more modes at high k. The number of modes grows as $k^{d-1}$.
Nevertheless, there is still no downward cascade for the special case of 2d fluid turbulence. The reason there is that there is a second continuous integral conserved quantity in this special case, the enstropy, which is the square of the curl of the 2d velocity.
$$ S = \int (\partial_y V_x - \partial_x V_y)^2 dx dy$$
This quantity has more derivatives than the energy, which is just $E=\int |v|^2$. The conservation laws require that the total $v_k$ and the total $k^2 |v_k|^2$ are both conserved, and if you just try to equipartition energy naively, you will increase the enstrophy by a huge amount, because the enstrophy of a high-k motion is just so much bigger. So you can't equipartition energy, you have to equipartition energy and enstropy together.
The law of enstropy partition requires paradoxically that the energy in the short modes is small. The enstropy equipartition is the important dynamics, and the energy ends up cascading the wrong way, from short wavelength to long wavelength modes, so that 2d fluids in a periodic box will cascade up to a single flow of two large counterrotating vortices.
The inverse cascade phenomenon was discovered in the 1960s, and it is most often attributed to Kraichnan. But several people noted the enstrophy cascade would wreck the traditional ideas for why turbulence occurs
The generic PDEs all have a turbulence, when they have a dissipation free regime, in a regime where the nonlinearity is important, but not the damping. This is studied today in models of preheating, in inflationary cosmology, and it is studied withing mathematics here and there.
The number of example systems is too large to list, it consists of any nonlinear nonintegrable equation without extra conserved quantities. Scale-invariant scalar field theory is a simple example, the one relevant for preheating:
$$ \partial_t^2 \phi_k - \nabla^2 \phi_k + \lambda_{ijlk}\phi_i\phi_j\phi_l = f(x,t) $$
With the appropriate choice of coefficients $\lambda$. You can also add mass scales, by adding a linear term in $\phi_k$, or additional quadratic terms which also come with an explicit scale. One is generally interested in the cascade at scales smaller than those defined by the low-order terms, so that the scale invariant cubic nonlinearity is the only important thing.
You should also add a damping, to give a short distance cut-off analogous to viscosity in turbulence, and you can do this by adding a $\partial_t \nabla^2\phi $ term with the appropriate coefficient, for example. I didn't do that, because in a numerical simulation, you can just do damping by artificially zeroing out very small k modes without an explicit local term to do this for you.
Strictly speaking, turbulence doesn't exist in two dimensions. The energy cascade required for turbulence to develop (transfer energy from large scales to small scales) is due to the (incompressible for illustration) vorticity equation:
$\frac{D\vec{\omega}}{Dt} = \left(\vec{\omega}\cdot\nabla\right)\vec{v} + \nu\nabla^2\vec{\omega}$
specifically the vortex stretching term:
$\left(\vec{\omega}\cdot\nabla\right)\vec{v}$
This term doesn't exist in two dimensional flow because $\vec{\omega} = \omega\hat{k}$ and $\nabla = \partial/\partial x + \partial/\partial y$ resulting in a zero dot product.
This stretching term is easily conceptualized by imagining an ice skater spinning. As the skater pulls in his/her arms, the rotation speed increases. Likewise, as a vortex tube gets elongated, the vorticity increases. It's this stretching of vortex tubes and the corresponding decrease in vortex radius and increase in rotational speed that causes large scale turbulence to cascade down to small scales where it becomes dissipated by the other term in the transport equation.
For a good explanation of the underlying physics of turbulence, read A First Course in Turbulence. It gives a very good understanding of the complex dynamics involved in both the generation of turbulence at the large scales and the cause of the cascade to progressively smaller scales.
As a hint, the non-linear convective terms in the Navier-Stokes equations are responsible for turbulence generation. It's easy to deduce why this is the responsible term by thinking through the role the other terms in the equation play by considering Kolmogorov's hypotheses.
Physically, I don't think "inward spiraly trajectory" is a good definition for turbulence. Random, 3D, chaotic fluid motion would be a better one.
Numerically, this means that the Navier-Stokes equations are a complex numerical system that do not accept simple analytical solutions for large values of the Reynolds numbers. Instead they display chaotic behavior such as sensitivity to initial conditions and a non-deterministic instantaneous solution (ie it is very difficult to know the exact state of the flow at a given point and instant). The simplified Lorenz attractor, loosely related to the NS equations, gives a good example of that behavior. It also shows that the solution is bounded (solution belongs to the strange attractor of fractal dimension between 2 and 3 in a 3D solution space) and that over time, we have a good idea as to where it lies. The analogous for the NS equations is that the statistics of turbulence are "relatively" easy to determine. Over time, we can compute the average behavior of the turbulent flow, we know universal behaviors exist locally (look up local isotropy and scale similarity) but it's still very hard to get an exact solution, to get the exact fluid trajectories you mentioned. Hopefully this gave a different perspective to your question than the previous excellent answer.
