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# Overview
* Theory
* Results from the literature:
* 2D Turbulence: Kraichnan (1967), Miller (1990), Robert & Sommeria (1992)
* Shallow water turbulence: Warn (1986), Renaud et al. (2016)
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# Theory
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## 2D Euler equation
$$\partial_t {\mathbf{u}} = - ({\mathbf{u}}.\nabla) {\mathbf{u}} - \frac{1}{\rho}\nabla p$$
Useful for studying 2D turbulence.
## Quasi geostrophic equation
Quasi-geostrophic equation conserves an approximate *potential vorticity*:
$$\frac{Dq}{Dt} = 0,$$
$$ q = \nabla^2 \psi + \frac{{f_0}^2}{\tilde \rho}
\left(\frac{\tilde \rho}{{N^2}} {\partial_z \psi} \right) + {\beta} y, $$
Incorporates rotation (Coriolis terms) and stratification (hydrostatic law) in
a 2D model. Bridging **ideal 2D turbulence** to **atmospheric turbulence**
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## 1-layer shallow water equations
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Fluid surface
![Shallow water](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./fig/swe_eta_h.png)
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Simple 2D model, useful for explaining geophysical phenomena.
$$\partial_t {\mathbf{u}} = - ({\mathbf{u}}.\nabla) {\mathbf{u}} -
g \nabla (h + h_b) -
f\mathbf{e_z} \times {\mathbf{u}} $$
$$\partial_t h = - \nabla \cdot (h {\mathbf{u}})$$
- where,
+ \\({\mathbf{u}}=\\) horizontal velocity vector,
+ \\(g =\\) acceleration due to gravity (related to wave speed),
+ \\(f =\\) Coriolis parameter (twice the angular velocity of Earth),
+ \\(h =\\) height of fluid
]
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## Properties of shallow water equations
- Invariants:
+ mass \\(h \mathbf{u} \\),
+ **energy** \\(E = E_K + E_P\\) (kinetic and potential energy)
+ **potential vorticity**, \\(q = (\omega + f)/h\\) and its higher powers
- Compared to **2D euler equations** or
**Quasi-Geostrophic equations** (Euler equations + coriolis force):
+ Admits gravity waves
- Velocity can be split using Helmholtz decomposition:
$${\bf u} = \bf{u}_r + \bf{u}_d$$
where,
+ \\({\bf u}_r = -\nabla \times ( {\bf e_z} \psi) \\) is the rotational component
+ \\(\bf {u}_d = \nabla \phi \\) is the divergent component
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# Results from literature
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## Common themes and relation between thermodynamic ensembles and geophysical flows
- Similar to:
- microcanonical ensemble which conserves (N, V, E) and
- canonical ensemble which conserves (N, V, T),
- two-dimensional models conserve (mass,
energy, vorticity / potential vorticity and its higher powers)
- Statistical mechanics used to theoretically explain:
- Existence of an equillibrium state in forced-dissipative flows
- Energy cascade between large and small scales
- Separation of scales: instead of continuum and molecules, between mean flow and turbulent fluctuations
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![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./fig/mesoscale.jpg)
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## Kraichnan (1967) - Inertial ranges in two-dimensional turbulence
- **2D turbulence** unlike Kolmogorov's **3D turbulence** theory:
**vorticity** and **enstrophy** conservation also plays a strong constraint
on cascade
- Constant enstrophy flux \\( \eta \\) would admit an extra inertial range.
Dual cascade:
$$E(k) \sim \epsilon^{2/3}k^{-5/3},\quad E(k) \sim \eta^{2/3}k^{-3}$$
- Cascade Directions
- Argued by by analysing transfer term which involves triad interactions of
wavenumbers \\( (k, p, q) \\)
$$ (\partial_t + 2\nu k^2) E(k,t) = \int \int T(k, p, q) dp dq$$
- \\(k^{-5/3}\\) range: constant energy flux, **inverse** cascade
- \\(k^{-3}\\) range: constant enstrophy flux, **forward** cascade
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## Bouchet & Vennaile (2012) - Statistical mechanics of 2D and geophysical flows (review paper)
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- **Miller (1990), Robert & Sommeria (1992)**: formulated equillibrium
statistical mechanics providing a Liouville theorem and expression for
entropy:
$$ S = \int d^2 r \int d\sigma \rho \ln \rho $$
where \\( \rho \\) is the pdf of the enstrophy \\( \sigma \\) within a
distance \\( r \\).
- **Van der Waals-Cahn-Hilliard model**.
- Describes phase transitions (equillibria of bubbles, soap films).
- Explanation for stability of the great red spot of Jupiter: a constant potential vorticity core, surrounded by
shear.
- A variational problem which minimizes free energy
]
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## Warn (1986) - Statistical mechanical equilibria of the shallow water equations
- Built upon ideas of Kraichnan (1967) etc. to include effects of small scale
inertio-gravity waves or *ageostrophic modes*.
- Considered quadratic constraints constructed from eigenmodes of the
linearised shallow-water system.
$$W = \\{ u, v, \eta \\}; \; W = \sum_{k,\alpha} A_k^{\alpha}(t) W_k^{\alpha}
\exp \left(i \mathbf{k} \cdot \mathbf{r} \right) $$
- Proof for Liouville theorem by expanding the triad interaction coefficients
in terms of eigenmodes, which in turn determines rate of change of the
coefficients \\( A_k^\alpha \\).
$$ A_k^\alpha = a_k^\alpha + i b_k^\alpha $$
$$
\begin{aligned}
\frac{\partial \dot{a}_k^\alpha }{\partial a_k^\alpha} +
\frac{\partial \dot{b}_k^\alpha }{\partial b_k^\alpha}
=2 \Re \frac{\partial \dot{A}_k^\alpha }{\partial A_k^\alpha}
=0
\end{aligned}
$$
- Studying the evolution of rotational \\( R \\) and inertio-gravity modes
\\( G \\) showed using multiple times scales:
- **Slow manifold**: For short time scales \\( \tau = \epsilon t \\), only surviving resonant
interactions are rotational and the motion is quasi-geostrophic
- **Forward energy cascade**: Evolution into a wave energy cascade and
equipartition spectrum requires extraction of energy from the rotational modes. Studied using a Langevin equation of \\(G\\).
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## Renaud et al. (2016) - Equilibrium Statistical Mechanics and Energy Partition for the Shallow Water Model
- Generalized and built upon previous studies in the context of shallow water
equations.
- Proved Liouville theorem for a triplet \\(h, hu, hv \\) and
transformed the result into \\(h, q, \mu \\)
- Using a discrete grid and a microcanonical ensemble of microstates in
\\( \\{ I_n, J_n, h_n, q_n, \mu_n \\}\\) a macrostate entropy was derived. In
the continuous limit:
![entropy](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./fig/swe_entropy.png)
in the absence of height variations / divergence it is
equivalent to Miller-Robert-Sommeria's form.
- By constructing the variational problem of entropy maximization under
constraints of conserved invariants:
![variational problem](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./fig/swe_var.png)
- Equipartition between kinetic and potential energy
- Study of two subsystems: mean flow and fluctuations as a coupled
variational problem
- Simplified variational problem on quasi-geostrophic pdf equivalent to
Miller-Robert-Sommeria's expression.
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# Thank you for your attention!
### Any questions?
![](https://cdn.jsdelivr.net/gh/ashwinvis/talks@3ebeb930f997b97dfcf87ce65fc5253a74eb9735/./images/noun_questions_inv.svg)
## slides will be uploaded: [fluid.quest/talks](https://fluid.quest/talks.html)
---
# References
1. Bouchet, F., and A. Venaille. “Statistical Mechanics of Two-Dimensional and Geophysical Flows.” Physics Reports 515, no. 5 (June 2012): 227–95. https://doi.org/10.1016/j.physrep.2012.02.001.
1. Chavanis, P. H., and J. Sommeria. “Statistical Mechanics of the Shallow Water System.” Physical Review E 65, no. 2 (January 14, 2002): 026302. https://doi.org/10.1103/PhysRevE.65.026302.
1. Kraichnan, R. H. “Inertial Ranges in Two-Dimensional Turbulence.” Phys. Fluids 10, no. 7 (1967): 1417–23. https://doi.org/10.1063/1.1762301.
1. Miller, J. “Statistical Mechanics of Euler Equations in Two Dimensions.” Physical Review Letters 65, no. 17 (October 22, 1990): 2137–40. https://doi.org/10.1103/PhysRevLett.65.2137.
1. Renaud, A., A. Venaille, and F. Bouchet. “Equilibrium Statistical Mechanics and Energy Partition for the Shallow Water Model.” Journal of Statistical Physics 163, no. 4 (May 2016): 784–843. https://doi.org/10.1007/s10955-016-1496-x.
1. Robert, R., and J. Sommeria. “Relaxation towards a Statistical Equilibrium State in Two-Dimensional Perfect Fluid Dynamics.” Physical Review Letters 69, no. 19 (November 9, 1992): 2776–79. https://doi.org/10.1103/PhysRevLett.69.2776.
1. Warn, T. “Statistical Mechanical Equilibria of the Shallow-Water Equations.” Tellus Ser. A-Dyn. Meteorol. Oceanol. 38, no. 1 (1986): 1–11.
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