# GEOF339

supert! Even Latex-maths seems to work!
$u_t + uu_x -fv = \dfrac{1}{\rho}\dfrac{\partial p}{\partial x}$

Maybe we should suggest this to Kjell Arild..
yeah, maybe
### Barotropic flow
With a constant density, there is **no vertical shear**. This follows from thermal wind:
$\dfrac{\partial}{\partial z} u_g = \dfrac{g}{\rho_0 f} \dfrac{\partial\rho}{\partial y} = 0$
$\dfrac{\partial}{\partial z} v_g = - \dfrac{g}{\rho_0 f} \dfrac{\partial\rho}{\partial x} = 0$
## SVERDRUP THEORY
*The vertical flow from the base of the surface Ekman layer, due to the windstress curl, drives a meridional motion.*
* The model is homogeneous in density and therfore completely ignore the dynamical effects of stratification
* Each of the oceans (The Atlantic, Pacific, etc. ) manifests the the westward intensification although the basins vary in shape, topography, stratification and the pattern of the winds.
* The Sverdrup relation shows that the fluid can only cross the isolines of *f* to the extend that vorticity is fed into the fluid column by the curl of the wind stress.
* If positive (negative) vorticty is imparted by a positive(negative) wind stress curl, the fluid must flow northwards(southwards).
* The Sverdrup solution will satisfy the BCs on both the eastern and western boundaries *only* if the total southward flux of the Sverdrup flow is zero. Since this is given in terms of the wind stress curl, only very special stress curls will satisfy.
* The Sverdrup equation cannot satisfy the BCs at both sides at the same time, nor can it satisfy the mass balance for the basin as a whole. Thus, there must be a region in the basin which does not follow the Sverdrup dynamics. The Sverdrup dynamics is heavily constrained by the $\beta$-effect (planetary vorticity gradient) which implies the existence of narrow intense boundary currents where the $\beta$-constraint is broken and the mass flux compensated. (The boundary currents depends only on the existence of $\beta$(the earths vorticity) and the fact that the ocean basins have boundaries. Not on the basin shape or the structure of the forcing)

## STOMMEL
Stommel realized that the Sverdrup relation needed additional dynamics to allow the return flow. The simplest way to do that is to add *linear bottom friction*. So below the surface Ekman layer, geostrophy is replaced by:
$-fv = -\dfrac{1}{\rho_c}\dfrac{\partial p}{\partial x} - ru$
$fu = -\dfrac{1}{\rho_c}\dfrac{\partial p}{\partial y} - rv$
where $(ru,rv)$ is a **frictional drag**. This breaks geostrophy, allowing for an ageostrophic return flow.
The modified Sverdrup relation then becomes:
$\beta v = \dfrac{1}{\rho_c H}\Big(\dfrac{\partial \tau^{wy}}{\partial x}-\dfrac{\partial \tau^{wx}}{\partial y}\Big)-r\Big(\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial y}\Big)$
Stommel then split the velocity into two parts, one interior one and one boundary one $v= v_I + v_B$.
##### The interior part:
$\beta v_I = \dfrac{1}{\rho_c H}\Big(\dfrac{\partial \tau^{wy}}{\partial x}-\dfrac{\partial \tau^{wx}}{\partial y}\Big)$

Assuming the wind stress curl is negative over the whole basin, the interior flow is southwards.

##### The boundary part
The return flow happens in a narrow boundary layer. The flow is strong, so this dominates the bottom drag:
$\beta v_B = -r\Big(\dfrac{\partial v_B}{\partial x}-\dfrac{\partial u_B}{\partial y}\Big)$
The meridional velocities are large in the boundary current and the boundary layer width is narrow, so we have:
$\beta v_B \approx -r\dfrac{\partial v_B}{\partial x}$
This has the solution:
$v_B = v_B(0)e^{-\beta x/r}$
which decays with increasing x.

If the boundary current is on the **western** side of the basin, $v_B$ **decays** into the interior. If the boundary is on the **eastern** side, the solution **grows** into the interior, which is **unphysical**.
(If the interior flow is northwards the boundary currents must go south. If the interior flow is southwards, the boundary currents must go north)

## QG- Dynamics

##### Motivation/concept
Here is a weather chart for Europe with isobars. If the flow follows the isobars exactly we would have geostrophic flow. Where the closer the isobar are the stronger the flow is. So geostrophic flow is a useful tool to explain the flow at a certain time, but it also means that the "isobar pattern" is "frozen" in time and can't change/move. In other words, geostrophy is limited as it is not timedependent and does not explain waves and topographic interactions.

In order to be able to make a prediction of the flow we will need to add advection and timedependency to the geostrophic flow. In **quasi geostrophy** we will look at small departures from geostrophy and seperate the divergent flow from the non-divergent flow.

If we are on a $f$-plane, where $f = f_0$ = constant, the geostrophic terms are strictly non-divergent and we can split the velocity terms into two. The geostrophic part (non-divergent) and the ageostrophic term (divergent).

But if $f$ varies, then the geostrophic terms will have a divergent part as well as the divergent ageostrophic term. So then the geostrophic streamfunction, which contains f, will not be a "proper" streamfunction. It will have distortions associated with the divergent geostrophic flow. So instead of seperating the geostrophic and geostrophic flow, we will seperate the non-divergent geostrophic part (which will be the same as if $f$=const.) from the rest (ageostrophic flow + divergent geostrophic flow). **The latter (rest) is small compared to the non-divergent geostrophic flow.**

Everything in QG-dynamics can be expressed by one variable, $\psi$.

Velocity:
$u = -\dfrac{\partial \psi}{\partial y}, v = \dfrac{\partial \psi}{\partial x}, w = \dfrac{f_0}{N^2}$

Pressure:
$p' = p_0 f_0 \psi$

Density:
$\rho = -\dfrac{\rho_0 f_0}{g}\dfrac{\partial\psi}{\partial z}$

### QG-dynamics application (Rossby waves)
QG-theory filters out fast moving gravity waves, but can describe the slow planetary waves that exists on the $\beta$-plane. To study these we will use the QGPV-equation.
$\dfrac{D}{Dt}\Big(\nabla^2\psi+\dfrac{{f_0}^2}{\partial^2\psi}{\partial z^2}\Big) + \beta \dfrac{\partial \psi}{\partial x} = 0$

For **Rossby waves** we will assume: No background flow $\rightarrow$ no advection. Then we will linearize.

$\dfrac{d}{dt}\Big(\nabla^2\psi+\dfrac{{f_0}^2}{\partial^2\psi}{\partial z^2}\Big) + \beta \dfrac{\partial \psi}{\partial x} = 0$

Here we can use $\dfrac{d}{dt}$ since we have no advection.

To solve the problem we will do these steps:
1. Assume a wave solution on the form: $\psi = \phi(z)\exp[i(kx+ly-\omega t)]$
2. Plug the solution in the QGPV equation, and get a 2nd order ODE for the amplitude $\phi$.
3. Solve the problem by using the BCs:
* no-slip w=0 at x= 0 $\Rightarrow \frac{d\phi(z)}{dz} = 0$ at z = 0
* At the surface the pressure is constant $p'=\rho_0 g\eta$ and $w = \frac{\partial\eta}{\partial t}$. Rewrite this in terms of the stream function

## Baroclinic modes

We use dynamic pressure and density, on the form:
$p = p_0 (z) + p’(x,y,z,t)$
$ρ = ρ_0 (z) + ρ’(x,y,z,t)$

We consider waves in a stratified ocean, where f = 0 and the Boussinesq approximation applies.
This allows us to simplify our equation of motion:

**Waves:**
- The fact that we're only interested in waves allows us to remove advection/mean flow from our eqs., since we are only interested in waves
**f = 0:**
- Allows us to remove Coriolis terms from eqs.

**Boussinesq:**
- allows us to treat ρ as $ρ_0$ in any term where it’s not multiplied with g.
- reduces the continuity equation to $\nabla(\vec{u}) = 0$

**Dynamic pressure and density:**
- $p_0$ and $ρ_0$ is in hydrostatic equilibrium by definition, and thus cancel eachother. We end up with in $p’$ and $ρ’$ in our vertical momentum equation
- $p_0 = p_0 (z)$ and $ρ_0 = ρ_0 (z)$ allows us to cancel them in horizontal derivatives. we end up with $p’$ and $ρ’$ in our equations in x-direction and y-direction.

The original momentum equation:

$$ \frac {D\vec{u}} {dt} + 2\vec{Ω} x \vec{u} - \vec{Ω}x(\vec{Ω}x\vec{r}) = -\frac1\rho* \dfrac{\partial p}{\partial z} + \vec{g} + \vec{F}
$$
When we make the above assumptions, and in addition igjonre friction and centrifugal acceleration, we obtain the wanted equations.

Step 1: obtaining the wave equation
a)
$\dfrac{\partial}{\partial x} (1) + \dfrac{\partial}{\partial y} (2)$
b) $\dfrac{\partial}{\partial t} (3)$ and (4) combined let's us include $N^2 = - \frac{g}{\rho_0} \dfrac{\partial\rho}{\partial z}$
c) d/dzdt and horizontal gradient squared gives us the wave equation

Step 2: consider ocean without boundaries
- Not interesting

Step 3: ocean with boundaries etha and -H
- --> wave channel / oceanic wave guide
- Wave signal trapped inside channel
- Reflected from boundaries

Step 4: apply to surface waves
Step 5: apply to internal waves --> here comes the modes
Step 6: do it all again for long waves