# Introduction to PO (Fluid Dynamics in the Ocean)

> Instructor: Ming-Huei Chang
> textbook: Introduction to Geophysical Fluid Dynamics, Benoit Cushman-Roisin and Jean-Marie Becker
>
Wave and turbulence: wind, solar heat, tide (gravity)
### Tidal force
Kind of body force due to the gravitation
Everywhere in the ocean is affected. Unlike solar radiation (only shallow surface)
- Surface wave
- inertial wave
- Rossby wave
- internal lee wave (in the interior of the ocean, internal stand for separation with surface wave)
- internal tide: the most common
## Current system
All the surfce current is derived by atmospheric wind (needed accurate atmospheric models)
NEC and STCC can grow mesoscale vortex.
## Newton’s second law
$$
\Sigma F = \frac{dp}{dt} = ma
$$
(rate of change of momentum)
The fluid is a particle in a flow field, unlike solid mechanism
$$
\begin{align}
dq/ dt &= \partial_t q + \partial_x q \partial_t x + \partial_y q \partial_t y + \partial_z q \partial_t z \\
&= \partial_t q + u\partial_x q + v\partial_y q + w\partial_z q
\end{align}
$$
### Inventory of forces in the ocean fluid
- Acceleration
- Advection
- PGF
- GF
- Acceleration associated with friction and viscosty (momentum flux = stress)
With friction and viscosity, the acceleration on the surface can affect deeper region.
- Coriolis Force $f = 2 \Omega \sin \psi$
> $\Omega =\frac{2\pi}{\textit{time of a revolution}}$
> $T = \frac{2\pi}{f} = \frac{\pi}{\Omega \sin \psi}$
### Balance between Inertial Acceleration and Coriolis Force
$$
\begin{align}
\frac{\partial u}{\partial t}-fv=0\\
\frac{\partial v}{\partial t}+fu=0
\end{align}
$$
### Momentum Equation in Geophysical Fluid Dynamics
$x$-momentum:
$$
\begin{align}
&\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}-fv\\
=&-\frac{1}{\rho_0}\frac{\partial p}{\partial x}+\frac{\partial}{\partial x}(\mathcal{A}\frac{\partial u}{\partial x})+\frac{\partial}{\partial y}(\mathcal{A}\frac{\partial u}{\partial y})+\frac{\partial}{\partial z}(\nu_E\frac{\partial u}{\partial z})
\end{align}
$$
$y$-momentum:
$$
\begin{align}
&\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}+fu\\
=&-\frac{1}{\rho_0}\frac{\partial p}{\partial y}+\frac{\partial}{\partial x}(\mathcal{A}\frac{\partial v}{\partial x})+\frac{\partial}{\partial y}(\mathcal{A}\frac{\partial v}{\partial y})+\frac{\partial}{\partial z}(\nu_E\frac{\partial v}{\partial z})
\end{align}
$$
$z$-momentum (Hydrostatic Approx.):
$$
0=-\frac{\partial p}{\partial z}-\rho g
$$
> $\frac{\partial v}{\partial t}$: Acceleration (local rate of change)
> $u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}$: Advection
> $fv$: Coriolis Force
> $\frac{1}{\rho_0}\frac{\partial p}{\partial y}$: PGF
> $\frac{\partial}{\partial x}(\mathcal{A}\frac{\partial v}{\partial x})+\frac{\partial}{\partial y}(\mathcal{A}\frac{\partial v}{\partial y})$: Horizontal Friction (x- and y-stress)
> $\frac{\partial}{\partial z}(\nu_E\frac{\partial v}{\partial z})$: Vertical Friction (z-stress)
> $\rho g$: Gravity Force
### Continuity Equation
$$
\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0
$$
Volumn is conserved. (Water is incompressible)
### Energy Budget
The equation governing temperature arises from conservation of energy.
$$
\rho C_v \frac{dT}{dt} + p \left( \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right) = k_T \nabla^2 T
$$
> LHS: pressure driven heat
> RHS: diffusion
> $k_T$: thermal conductivity
>
Simplify with Continuity Equation:
$$
\rho C_v \frac{dT}{dt} = k_T \nabla^2 T
$$
With Newton's Second Law:
$$
\frac{dT}{dt}=\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z} = \kappa_T \nabla^2 T,\ \kappa_T\frac{k_T}{\rho_0C_v}
$$
> $\kappa_T$: heat kinematic diffusivity (which is much larger than $\kappa_S$)
### Salt Budget
Density varies with salinity.
Seawater parcel conserved in salt content (if diffusion DNE)
$$
\frac{dS}{dt} = \frac{\partial S}{\partial t}+u\frac{\partial S}{\partial x}+v\frac{\partial S}{\partial y}+w\frac{\partial S}{\partial z} = \kappa_S \nabla^2 S
$$
> $\kappa_S$: coe of salt diffusion

### EOS
Relation between pressure and density (related to pressure, temperature, salinity).
$$
\rho = \rho_0 [1-\alpha(T-T_0)+\beta(S-S_0)]
$$
> $\alpha$: coe of thermal expansion
> $\beta$: coe of saline contraction

If diffusion is governed by molecular process, $\kappa_S$ is equals to $\kappa_T$.
Small-scale (~ meter): molecular diffusion
Large-scale: turbulet diffusion / eddy diffusivity (eg. eddy)
$$
\begin{align}
&\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}+v\frac{\partial \rho}{\partial y}+w\frac{\partial \rho}{\partial z}\\
=&\frac{\partial}{\partial x}(\mathcal{A}\frac{\partial \rho}{\partial x})+\frac{\partial}{\partial y}(\mathcal{A}\frac{\partial \rho}{\partial y})+\frac{\partial}{\partial z}(\kappa_E\frac{\partial \rho}{\partial z})
\end{align}
$$
> $\frac{\partial \rho}{\partial t}$: local rate of change of density
> $u\frac{\partial \rho}{\partial x}+v\frac{\partial \rho}{\partial y}+w\frac{\partial \rho}{\partial z}$: Advection
> $\frac{\partial}{\partial x}(\mathcal{A}\frac{\partial \rho}{\partial x})+\frac{\partial}{\partial y}(\mathcal{A}\frac{\partial \rho}{\partial y})$: Horizontal Diffusion ($\mathcal{A}$: horizontal eddy diffusivity)
> $\frac{\partial}{\partial z}(\kappa_E\frac{\partial \rho}{\partial z})$: Vertical Diffusion ($\kappa_E$: vertical eddy diffusivity)

## Eddy diffusivity
As numerical model is limited in spatial resolution:
**Subgrid-scale parameterization**: how unresolved turbulent and subgrid-scale motions affect larger resolved flow.
**Dissipation and Mixing**: primary effect subgrid-scale motion and turnulence (small eddies, billows).Represent the motion and Reynold's stress with **super viscosity**.