# Few observations in the atmosphere and the oceans

## Measurements in the atmosphere

The velocity and temperature in the free atmosphere (high troposphere,
stratosphere and mesosphere) can be measured by probes fixed on aircrafts, ballons
or rockets, giving one-dimensional measurements. There are also LIDAR and RADAR
measurements. Globally, the measurements are very sparsed.

### Horizontal profiles

From measurements of probes on aircrafts, one can compute signals as a function of
an horizontal coordinate (Taylor approximation). The simplest quantity than one
can compute from these signals are "horizontal spectra". Figure
{numref}`fig_NastromGage1985` show 3 spectra of zonal wind, meridional wind and
potential temperature.

```{figure} ./figs/fig_NastromGage1985.png
---
width: 70%
name: fig_NastromGage1985
---
From left to right, variance power spectra of zonal wind, meridional wind, and
potential temperature near the tropopause from Global Atmospheric Sampling
Program aircraft data. The spectra for meridional wind and temperature are
shifted 1 and 2 decades to the right, respectively. Reproduced from
{cite:t}`NastromGage1985`.
```

The ${k_h}^{-5/3}$ scaling low suggests an energy cascade but it cannot be related
to standard isotropic turbulence since (i) the atmosphere is very shallow and (ii)
the flows at such large scales are strongly influenced by rotation and
stratification.
We will present different interpretations of these spectra.

Observations indicate that the energy cascade is downscale at scales below 100 km
(Lindborg & Cho 2001).

We will see that some decompositions (horizontal vorticity/horizontal divergence,
geostrophic/ageostrophic) can be useful... But controversy (incompatible results):

- {cite:t}`Lindborg2007Horizontal`
- {cite:t}`callies2014transition`
- {cite:t}`Lindborg_2014`
- {cite:t}`WangBuhler2019` (see also {cite:t}`Waite_2020`)

### Vertical profiles

${k_z}^{-2}$ at the largest scales, ${k_z}^{-3}$ at intermediate scales and
shallowing towards ${k_z}^{-5/3}$ at smallest scale. See for example
{cite:t}`FrittsAlexander2003`.

## Measurements in the oceans

Garrett & Munk (1975) produced an analytical models able to approximatelly
describe oceanic spectra computed from measurements. Temporal and vertical
spectra measurements are very common (fixed and dropped probes, respectivelly).
Horizontal spectra are more difficult to measure (towed probes).

For the temporal spectra, the energy is concentrated at frequencies between $f$
and $N$, indicating that internal gravity waves are important.

Garrett & Munk model assumes that the spectra are only due to waves.

TODO: Lvov...

### Vertical profiles

```{figure} ./figs/fig_gargett1981.png
---
width: 70%
name: fig_gargett1981.png
---
Vertical wavenumber spectra of vertical shear in the ocean. The spectra are
normalized by ... and the wavenumber is normalized by $k_o$. Reproduced from
{cite:t}`Gargett_1981`.
```

### Horizontal profiles

```{figure} ./figs/fig_KlymakMoum2007b.png
---
width: 70%
name: fig_KlymakMoum2007b.png
---
Average isopycnal slope spectra, binned in one-decade bins of the turbulent
diffusivity ... and normalized by $N_0/N$ (from {cite:t}`KlymakMoum2007`).
A line with slope $k_x^{1/3}$ has been added for comparison.
Taken from {cite:t}`RileyLindborg2008`.
```

```{figure} ./figs/fig_Ewart1976.png
---
width: 50%
name: fig_Ewart1976.png
---
Power spectra of temperature vs horizontal wavenumber $\alpha = k_h$
computed from runs off the coast of San Diego (30°N, 124°W).
The spectra are normalized by $N/N_0G$, where $G$ is the temperature
gradient averaged over several measurements, and the buoyancy frequency $N$
has been fitted to the exponential profile $N = N_0 e^{z/b}$. The units of
the spectra are m$^2$/cpm. Reproduced from {cite:t}`Ewart1976`.
```

## Summary

- Horizontal spectra steep (${k_h}^{-3}$) at large scale and ${k_h}^{-5/3}$ at
  moderate scales ($\sim$ 100 km in the atmosphere and $\sim$ 5 km in the oceans).

- Downscale energy cascade (Lindborg (1999); Cho & Lindborg (2001); Lindborg & Cho
  (2001)).

- ${k_z}^{-2}$ at the largest scales, ${k_z}^{-3}$ at intermediate vertical scales
  and shallowing towards ${k_z}^{-5/3}$ at smallest scale.

- Still contreverse about decomposition... Dominated by ageostrophic modes.

## Different interpretations of the geophysical spectra

### Two "old" hypothesis

- Waves turbulence ({cite:t}`Dewan1979` for the atmosphere, Garrett & Munk 1975
  for the oceans): downscale energy cascade

- Stratified turbulence (Lilly 1983): upscale (inverse) energy cascade, similar to
  2D turbulence, like quasi-geostrophique limit (potential enstrophy conserved and
  proportional to the energy in Fourier space).

Hopefully, these two very different interpretations lead to opposite predictions:
wave theories predict a forward cascade, while vortex Lilly’s theory predicts a
backward cascade.

The direction of the energy cascade in the atmosphere has been measured by
Lindborg (1999); Cho & Lindborg (2001); Lindborg & Cho (2001). The third order
structure functions are shown to be proportional to negative $r$, where $r$ is the
separation distance, which implies a downscale energy cascade. This tends to show
that the energy spectra can not be explained by a two-dimensional dynamics and
thus gives credit to the wave theories that predict a downscale energy cascade.

### More theories based on waves

The concept of "wave saturation" has been used to explain the universality of the
spectra (see Fritts, 1984, for a review). All waves are marginally stable, i.e.
nearly “breaking”, which fixes the energy distribution. The physical explanation
of the saturation is however unclear: Dewan & Good (1986) and Smith et al. (1987)
invoke linear instabilities while Hines (1991) and Dewan (1997) proposed
non-linear processes. In fact, it is known that most of the waves are strongly
non-linear in the oceans and atmosphere ({cite:t}`FrittsAlexander2003`; Lindborg &
Riley, 2007).

- Weak wave turbulence

- Wave scattering

### More theories based on vortices

- QG vortices (Tulloch & Smith 2009), surface quasigeostrophy

- stratified turbulence (Lindborg 2006), nonlinear anisotropic turbulence