A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czir ok et al. 1995, describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are dissussed and simulation of the ODEs presented.

The existence and uniqueness of solutions of the Navier-Stokes equation driven with additive noise in three dimensions is proven, in the presence of a strong uni-directional mean flow with some rotation. The physical relevance of this solution and its relation to the classical solution, whose existence and uniqueness is also proven, is explained. The existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated -5/3 power law for the decay of the power spectrum of a turbulent 3-dimensional flow.

Recent advances in the theory of turbulent solutions of the Navier-Stokes equations are discussed and the existence of their associated invariant measures. The statistical theory given by the invariant measures is described and associated with historically-known scaling laws. These are Hack's law in one dimension, the Bachelor-Kraichnan law in two dimensions and the Kolmogorov's scaling law in three dimensions. Applications to problems in turbulence are discussed and applications to Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES) models in computational turbulence.

Kolmogorov's statistical theory of turbulence is based on the existence of the invariant measure of the Navier-Stokes flow. Recently the existence of the invariant measure was established in the three-dimensional case. It was established earlier by the author for uni-directional flow and for rivers. We discuss how one can try to go about approximating the invariant measure in three dimensions.

The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one dimensional turbulence is established. The turbulent solutions are not smooth but H\"older continuous with exponent $3/4$. The scaling of the solutions' second structure (or width) function gives rise to Hack's law \cite{H57}; stating that the length of the main river, in mature river basins, scales with the area of the basin $l \sim A^{h}$, $h = 0.568$ being Hack's exponent.

In this paper, we will build mathematical pelagic fish migration models and inspect simulations to discover the behavior for motion amongst a school. I focus on the Icelandic Capelin, and explore all possible migration patterns and their limitations as governed by mathematical equations and not biological observations. We develop a system of ordinary differential equations from a discrete system for the most general motion. First, we will perturb the system to develop a system of stochastic differential equations to study the unique behavior under naturally occurring external forces such as currents or reefs. Then, we will categorize the found transitory and long term behavior of these systems and compare them to the unperturbed solutions. Finally, we will prove that the system exhibits a subcritical pitchfork bifurcation.

We compare the predictions of the stochastic closure theory (SCT) [1] with experimental

data obtained in the Variable Density Turbulence Tunnel (VDTT) [2], at the Max Planck Institute

for Dynamics and Self-Organization in G¨ottingen. The mean flow in the homogeneous

turbulence experiment reduces the number of parameters in SCT to just three, one characterizing

the variance of the mean field noise and another characterizing the rate in the large

deviations of the mean. The third parameter is the decay exponent of the Fourier variables in

the Fourier expansion of the noise. This characterizes the smoothness of the turbulent velocity.

We compare the data for the even-order longitudinal structure functions ranging from the

the second to the eighth structure function as well as the third-order structure function, to

the SCT theory with generic noise, depending on the above three parameters, at five Taylor-

Reynolds (Rl) numbers ranging from 110 to 1450. The theory gives excellent comparisons

with data for all the structure functions and for all the Taylor-Reynolds numbers. This highlights

the advantage of the SCT theory, where the structure functions can be computed explicitly

and their dependence on the Rl number computed. These results are robust with respect

to the size of dissipation range filters applied to the data, and comparisons to the fits without

the Rl number corrections show a clear improvement when the corrections are present. This

improvement is significant for the lower Rl number and disappears as the Rl number becomes

large, as expected. Very surprisingly the comparison of SCT and the data also gives information about the

smoothness of the turbulent velocity as Rl becomes very large.

We then compare the SCT to the Townsend-Perry constants generated in the flow physics

facility (FPF) at the University of New Hampshire. The Reynolds correction terms in the SCT

changed the initial derivation of this similarity, forcing a refinement of the theory. Once done,

we see good agreement between the data and the SCT.