The Water Column¶

import xarray as xr
from matplotlib import pyplot as plt
import numpy as np
import gsw
%matplotlib inline
plt.rcParams['figure.figsize'] = (12,7)
plt.rcParams['font.size'] = 16
plt.rcParams['lines.linewidth'] = 2
from IPython.display import SVG, display, Image, display_svg
import holoviews as hv
hv.extension('bokeh')

ARGO Floats¶

Content from https://argo.ucsd.edu/how-do-floats-work/

argo float

argo cycle

Temperature - Salinity Relationship¶

Used to identify water mass properties.

fig, ax = plt.subplots(figsize=(9,7))
plt.plot(argo.s_a, argo.c_t, c='0.7', zorder=-999)
scat = ax.scatter(argo.s_a, argo.c_t, c=argo.pres)
ax.set_xlabel(r'$S_A$ (g/kg)')
ax.set_ylabel(r'$\Theta$ $^\circ$ C')
plt.colorbar(scat, label='pressure [dbar]')
plt.grid()
plt.close()

fig

Buoyancy¶

A water parcel experiences a buoyancy force if its density differs from the ambient density.

buoyancy

Static Stability¶

Static stability measures how quickly a water parcel is restored to its position in the water column if it is displaced vertically. If the stability is negative, the water column has the potential to overturn.

An obvious, but slightly flawed, measure of stability would be

\[ -\frac{g}{\rho} \frac{\partial \rho}{\partial z} \]

Question: What is wrong with this definition?

Density Profile¶

In-situ density $\rho(S_A, \Theta, p)$ is not actually very informative about buoyacy because it mostly shows the adiabatic compressibility of seawater.

fig, ax = plt.subplots()
argo_p.rho.plot(y='pres', yincrease=False, ax=ax)
ax.set_ylim([2000, 0])
ax.set_ylabel(r'$p$ (dbar)')
ax.set_xlabel(r'density (kg m$^{-3}$)')
plt.close()

fig

Buoyancy Frequency¶

The problem with the previous definition of stability is that it doesn’t account for the adiabatic compressibility of water. If a parcel is displaced vertically, its pressure will change, causing a change in density. We need to subtract this effect.

\[ N^2 = -g \left ( \frac{1}{\rho}\frac{\partial \rho}{\partial z} - \kappa \frac{\partial p}{\partial z} \right ) = g \left (\alpha^\Theta \frac{\partial \Theta}{\partial z} - \beta^\Theta \frac{\partial S_A}{\partial z} \right ) \]

$N$ is called the buoyancy frequency because a displaced parcel will oscillated with frequency $N$. This type of oscillation is called an internal wave. If $N^2$ is negative, the water column is unstable and will overturn.

fig, ax = plt.subplots()
ax.plot(Nsquared**0.5, p_mid)
ax.set_xlabel(r's$^{-2}$')
ax.set_title('Buoyancy Frequency')
ax.vlines(60**-1, 2000, 0, label='1 minute', color='k')
ax.vlines((60*60)**-1, 2000, 0, label='1 hour', colors='k')
ax.set_ylim([2000, 0])
ax.set_ylabel(r'$p$ (dbar)')
ax.set_xlabel(r'$N$ (s$^{-1}$)')
plt.close()

/tmp/ipykernel_1599/960056500.py:2: RuntimeWarning: invalid value encountered in sqrt
  ax.plot(Nsquared**0.5, p_mid)

fig

Potential Density¶

To eliminate the effect of adiabatic compressibility, we can calculate the denisty the water would have if brought to some reference pressure $p_{ref}$. This is called potential density, defined as

\[ \sigma = \rho(S_A, \Theta, p_{ref}) - 1000 \]

Because $\sigma$ is only a function of quasi-conservative quantities ($S_A$ and $\Theta$) it is itself quasi-conservative.

Near the reference pressure, it is the case that $$ N^2 \simeq -\frac{g}{\rho} \frac{\partial \sigma}{\partial z} $$ but, due to the nonlinearity in the equation of state, this is not true far from the reference pressure.

It is common to denote the reference pressure using a subscript: $\sigma_0$ (0 dbar), $\sigma_2$ (2000 dbar), and $\sigma_4$ (4000 dbar).

fig, ax = plt.subplots()
(argo_p.rho-1000).plot(y='pres', yincrease=False, label=r'$\rho - 1000$')
argo_p.sig0.plot(y='pres', yincrease=False, label=r'$\sigma_0$')
argo_p.sig2.plot(y='pres', yincrease=False, label=r'$\sigma_2$')
argo_p.sig4.plot(y='pres', yincrease=False, label=r'$\sigma_4$')
ax.set_ylim([2000, 0])
ax.set_ylabel(r'$p$ (dbar)')
ax.set_xlabel(r'density (kg m$^{-3}$)')
ax.legend(frameon=True, loc='lower right')
plt.close()

fig

fig, ax = plt.subplots()
(argo_p.rho-argo_p.rho[0]).plot(y='pres', yincrease=False, label=r'$\Delta \rho$')
(argo_p.sig0-argo_p.sig0[0]).plot(y='pres', yincrease=False, label=r'$\Delta \sigma_0$')
(argo_p.sig2-argo_p.sig2[0]).plot(y='pres', yincrease=False, label=r'$\Delta \sigma_2$')
(argo_p.sig4-argo_p.sig4[0]).plot(y='pres', yincrease=False, label=r'$\Delta \sigma_4$')
ax.set_ylim([2000, 0])
ax.set_ylabel(r'$p$ (dbar)')
ax.set_xlabel(r'density difference (kg m$^{-3}$)')
ax.legend(frameon=True, loc='upper right')
plt.close()

fig

Intro to Physical Oceanography