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\chapter{Two-dimensional models of geophysical turbulence}\label{ch:simul}
Atmospheric turbulence is characterized by a wide range of scales, ranging from
the order of millimetres to several thousands of kilometres. Understanding the
underlying interactions behind the energetics is essential to improve general
circulation models (GCM) while also recognizing the limits of predictability
[see @lorenz_predictability_1969; @vallis_atmospheric_2017 pp. 433--447]. As a
general rule of thumb, as our capabilities of modelling of the small scales
improve, so does the predictability, which has motivated researchers
to advance our understanding of geophysical turbulence.
The scaling laws for the energy distribution in the atmosphere were revealed
through the wind and temperature measurements made in Global Atmospheric
Sampling Program (GASP, see @nastrom_kinetic_1984 and @NastromGage1985). In
GASP, data spanning across the globe was collected using over 6900 commercial
flights during the years 1975-79. At least 80% of the data were measured between
altitudes of 9 and 14 kilometres, near the tropopause. The spectra thus
calculated, shown in
@fig:nastromgage, revealed that there are two separate ranges: in a
narrow synoptic range (wavelengths between 1000 and 3000 km) the energy
spectrum scales as
$k^{-3}$, and in the mesoscale range (wavelengths between 2 and 500 km) it
scales
as \kfivethird. For wavelengths just above 500 km there is also an overlap
region where these two ranges blend.
Prior to this study the $k^{-3}$ scaling was well-known [@Charney1971] and
explained using theories for two-dimensional turbulence
[@Kraichnan1967]. However the theoretical explanation of the \kfivethird\ mesoscale
spectra and the direction of the energy cascade has remained under debate
for several decades [@kitamura_energy_2010; @vallgren_possible_2011-1]. Two
hypotheses -- one suggesting a forward energy cascade [@Dewan:1979] and another
suggesting an inverse energy cascade [@Gage:1979; @Lilly:1983] have been at the
heart of this debate. Alternative hypotheses were also introduced in the later
years which will be described in the forthcoming sections. In
@sec:energy, we will also see why there is an emerging consensus that
the mesoscale range is the result of a forward cascade of energy into
smaller scales [@Lindborg1999;@ChoLindborg2001].
{#fig:nastromgage width=70%}
Of particular relevance for this thesis is the theoretical prediction,
estimating the vertical resolution required for reproducing the mesoscale
spectra. In @Lindborg2006 and @Waite-Bartello:2004 the vertical resolution was
estimated from the vertical length scale of the elongated structures in
stratified turbulence, i.e. $l_v \sim u/N \approx \order{1} \text{km}$. The
hypothesis of @Callies-Buhler-Ferrari:2016 implies that an even finer
resolution, of \order{0.1} km, would be needed to resolve inertia-gravity waves and
thus the mesoscale motions. Contrary to these expectations, in @AugierLindborg2013,
some GCM runs reproduced the energy spectrum at mesoscales, using a coarse
vertical resolution of only 24 pressure levels. Additionally, the spectral
energy budget calculations in @AugierLindborg2013 exhibited a forward
energy cascade in the mesoscale range. This result prompted the question, why this was
possible despite the theories predicting a finer resolution requirement. In
trying to answer this question, we have simulated the most extreme case: a
quasi-two dimensional model which accommodates both waves and vortices,
equivalent to simulating a single layer of fluid.
The first candidate for this exercise was the classical shallow water
equations. Large scale wave forcing in a narrow band of wavenumbers (forcing
wavenumber, $k_f = 6\delta k$) was used to excite the flow. Consistent with the
expectations, shallow water dynamics exhibited a forward cascade. However,
the waves simulated using the shallow-water equations tend to coalesce to form
shock fronts. The resulting cascade was weaker
than what is observed at
mesoscales and the spectrum scaled as $k^{-2}$. Nevertheless, the dynamics of
shock-dominated wave turbulence caught our interest and we derived novel
scaling theories for spectra, shock separation, structure
functions and other related quantities. These results are
presented in @augier_shallow_2019. The study can be potentially extended to
other domains such as acoustics, but is unlikely to find straightforward
applications in geophysical turbulence. Therefore in a second study
[@LindborgMohanan2017] we modified the shallow water equations into a toy
model, which does not cause waves to evolve into shocks and has nice properties
such as a quadratic expression for kinetic energy. Another advantage of using
the toy-model is that we obtain a $k^{-5/3}$ spectrum and similar
dynamics, as seen in the GCM reported in @AugierLindborg2013.
In the first section in this chapter, we present a background of the theoretical,
experimental and computational attempts towards understanding the mesoscale energy
cascade. The next section describes some properties of the shallow water and
toy model equations. The subsequent sections showcase some interesting results
from @augier_shallow_2019 and @LindborgMohanan2017.
Background
Two-dimensional turbulence
The latter half of the twentieth century presented exciting insights into how
kinetic energy is distributed among different scales in the atmosphere.
Several researchers in the 1960s found that energy spectra associated with
synoptic scales of motion ($\sim 1000-3000$) km are very different from
spectra of small scale three-dimensional isotropic turbulence. Through
observational evidence [@horn_analysis_1963] and later on by GCM
calculations [@wellck_effect_1971], it was found that the energy spectrum
scales as $k^{-3}$ at synoptic scales.
Therefore, the underlying mechanism is different in comparison with 3D
isotropic turbulence, in which energy spectrum scales as $k^{-5/3}$.
It was also realized that in two-dimensional turbulence, vorticity and
enstrophy conservation places a
strong constraint on the cascade [see for example, @fjortoft_changes_1953].
This development led to the seminal work by @Kraichnan1967 wherein a theory
for a coexistence of a dual cascade was formulated. A
large scale inverse-energy cascade was predicted
wherein, the energy spectrum scales as $E(k) \sim \epsilon^{2/3} k^{-5/3}$,
where $\epsilon$ is the energy flux through the cascade. At smaller scales there is a
forward enstrophy cascade with an associated spectrum, $E(k) \sim \eta^{2/3}
k^{-3}$, where $\eta$ is the enstrophy flux. One way to deduce these
power laws was to invoke similar assumptions as @Kolmogorov1941. It was assumed
that the \kfivethird\ inertial range solely depends on wavenumber $k$ and
$\epsilon$, and likewise the $k^{-3}$ range would
depend on $k$ and $\eta$. A more formal
approach relying on statistical mechanics arguments was also formulated by
@Kraichnan1967 to arrive at the same conclusion and predict the direction of
the cascades.
Kraichnan studied how triad interactions would act in
two dimensions, using the incompressible Navier-Stokes equations, which
have two quadratic inviscid invariants, energy and enstrophy. The spectral
energy and enstrophy fluxes can be expressed as,
\begin{align}
\Pi(k) &=
\frac{1}{2} \int_0^k dk' \int dp \int dq T(k', p, q) -
\frac{1}{2} \int_{k}^\infty dk' \int dp \int dq T(k', p, q),
\
Z(k) &=
\frac{1}{2} \int_0^k k'^2 dk' \int dp \int dq T(k', p, q) -
\frac{1}{2} \int_k^\infty k'^2 dk' \int dp \int dq T(k', p, q),
\end{align}
where $T$ is the energy transfer function arising from the nonlinear term in
the Navier-Stokes equations. The fluxes were found to arise from two classes of
mutually exclusive interactions, the range $k' \in [k, \infty)$ would interact
with all wavenumbers $p, q < k$ and similarly the range $k' \in [0, k]$
would interact with all wavenumbers $p, q > k$. Using this as a starting point,
it was shown that a constant energy flux $\Pi(k)$ is obtained where the
energy spectrum scales as \kfivethird\ as well as a constant enstrophy flux
$Z(k)$ where the energy spectrum scales as $k^{-3}$. The directions of cascades,
i.e. the signs of the fluxes used in scaling the inertial ranges, were then
determined using statistical mechanics arguments [see also
@kraichnan_two-dimensional_1980].
Quasi-Geostrophic equations
Despite the firm foundations that the theory of @Kraichnan1967 presented, a gap
left to be bridged -- to connect the ideal two-dimensional turbulence to
atmospheric turbulence. @Charney1971 pondered if it was possible to realize the
predictions at all and if so, within what limits the atmosphere can be considered
two-dimensional. It is well-known that most chaotic motions at planetary scales
originate from baroclinic instability [@vallis_atmospheric_2017]. The effects of rotation and
stratification were not considered in @Kraichnan1967. These "shortcomings" were
addressed to some extent by @Charney1971, who derived the $k^{-3}$ spectrum by
analysing the so-called quasi-geostrophic (QG) equation, conserving an
approximate expression for potential vorticity:
$$\Dt{q} = 0,$$
$$ q = \nabla^2 \psi + \frac{f_0^2}{\tilde \rho}\left(\frac{\tilde \rho}{N^2}
\p_z \psi \right) + \beta y, $${#eq:quasigeo}
where $\psi$ is the horizontal stream function, $f_0$ is the solid body rotation
speed of the frame of reference, $\tilde \rho$ is the potential density, $N$ is
the Brunt-\text{V"ais"al"a} frequency and $\beta \approx \p_y f$ is the beta
parameter. This equation is valid when certain criteria are met:
Strong rotation, implying that the Rossby number, $Ro = U/f_0L$, is much
smaller than unity, where $U$ and $L$ are characteristic horizontal
velocity and length scales, respectively.
Characteristic length scales of motion are of the same order as Rossby radius
of deformation, i.e. $L \sim L_d = HN/f_0$ or $Ro ({L}/{L_d})^2 =
\order{Ro}$. Here, $H$ is a characteristic vertical length scale.
Variations in the Coriolis force ($\beta$) are small, implying scales may not
be as large as planetary length scales.
\noindent Apart from these criteria there are some other scale restrictions [see
chapter 5 in
@vallis_atmospheric_2017]. Using @eq:quasigeo and the result that both energy
and QG enstrophy ($q^2 / 2$) are conserved quantities it was shown that
the forward energy cascade can be inhibited by the geostrophic constraint. In
this respect the QG equation behave similar to the 2D Navier-Stokes equations.
The $k^{-3}$ scaling law was derived for the QG equations in @Charney1971 and
GCM results from @wellck_effect_1971 were used to confirm the existence of the
$k^{-3}$ spectrum.
Energy cascade in synoptic and mesoscale flows{#sec:energy}
{#fig:cascade width=70%}
Nowadays, it is understood that the $k^{-3}$ spectrum at the synoptic scales
(typically, for wavelengths over a thousand kilometres) is an example of a
Kraichnan-Charney type of turbulence, with enstrophy cascading
downscale. The top plot of @fig:cascade shows how @Kraichnan1967 anticipated
the two scaling laws would coexist -- a "stirring force" would inject
energy at intermediate scales, which would then cascade towards
small wavenumbers, while enstrophy would cascade in the opposite direction. In
contrast to this picture, the study by @NastromGage1985
revealed spectra that were similar to the sketch on the bottom plot of
@fig:cascade.
Here, the synoptic scale $k^{-3}$-range is found at larger scales than the
mesoscale $k^{-5/3}$-range, an observation that @Frisch found 'paradoxical'.
By what the mechanism the \kfivethird\ mesoscale spectrum is produced has been
an open question ever since, and competing theories have been put forward to address
this issue. @Dewan:1979 analysed the energy spectrum of velocity fluctuations in the
stratosphere up to wavelengths of \order{10} kilometres and suggested that
internal gravity waves, feeding on turbulent layers trapped by large scale
shear flows, could be the driving mechanism behind the spectrum.
He considered the mesoscale spectrum to be analogous to the ocean
spectrum reported by @garrett_space-time_1972. It was also asserted that a
Kolmogorov-type of forward energy cascade of waves is present. This
was substantiated using a simple model for a shear flow by @Phillips.
In @Gage:1979, a competing hypothesis was formulated assuming that the mesoscale cascade
process would be similar to Kraichnan's prediction of an inverse energy cascade.
To confirm the power law, the two-point temporal structure function of winds
was derived and applied on data from balloon sounding measurements. Through
Taylor's transformation the author linked the temporal variability to the
spatial structure function equivalent to a \kfivethird\ spectrum. In a
contemporary paper,
@Lilly:1983 also made a similar conjecture of an inverse energy cascade, in
studying decaying stably-stratified turbulence and its tendency to evolve into
enlarged vortices. A scale based decomposition of the Boussinesq equation into
waves and vortices by @riley_direct_1981, was used to study the interactions of
waves and turbulence, using an initial state of homogenous and isotropic
turbulence. It was hypothesized that the stratified turbulence would
transfer energy to larger scales, resulting in a $\kfivethird$
spectrum. The inverse energy cascade hypothesis was revisited by @Xia2011
through experiments, wherein a large scale planar vortex was forced from the
small scales electromagnetically to generate a \kfivethird\ spectrum.
@Lindborg1999 contains a detailed review of the above and other hypotheses,
proposed in the light of the @nastrom_kinetic_1984 results, along with a
discussion of their pros and cons. The two-dimensional turbulence
interpretation can be questioned since it would require a small-scale energy
source at \order{1} km, where the mesoscale to microscale transition is observed
[@vinnichenko_kinetic_1970]. However, at this scale three-dimensional motions
are dominant.
To reconcile the inverse energy cascade hypothesis with the observed
transition from a $k^{-5/3}$ to a $k^{-3}$ spectrum, @gage_theoretical_1986
introduced the hypothesis that there is an energy sink at the transition
scale. However, it is unclear what the physical mechanism would be that
could act as such a sink.
Instead of applying a spectral analysis of aircraft data
@Lindborg1999 carried out a structure function analysis. A
velocity structure function is the statistical moment the velocity
difference
$\delta {\bf u} = {\bf u}({\bf x} + {\bf r} ) - {\bf u}( {\bf x} )$.
In particular, the second order longitudinal and transverse
structure functions are defined as $\langle \delta u_L ^2 \rangle$
and $\langle \delta u_T^2 \rangle$, where $u_L$ is the velocity
component in the 'longitudinal' direction, that is the same
direction as ${\bf r}$, $u_T$ the velocity component in a direction
which is perpendicular to ${\bf r}$, and $\langle \rangle$ is a mean
value. If the separation vector is horizontal the structure
functions can be assumed to be independent of position ${\bf x}$
(statistical homogeneity) and independent of the direction of
${\bf r}$ (statistical isotropy), thus being functions only of
$r = | {\bf r} |$. Under the assumptions of homogeneity and isotropy,
@Lindborg1999 derived the two-dimensional counterparts of the so
called four-fifths law for the third order structure function
[@Kolmogorov1941] of three dimensional turbulence
$$
\langle \delta u_L^3 \rangle = - \frac{4}{5} \epsilon r , ,
$${#eq:K41}
where $\epsilon$ is the downscale energy flux, which is also equal to the
energy dissipation. In two-dimensions there are two third order structure
function laws. In the inverse energy cascade range we have
$$
\langle \delta u_L^3 \rangle = \frac{3}{2} P r , ,
$${#eq:2DF}
where $P$ is the (positive) upscale energy flux, and in the enstrophy cascade
range we have
$$
\langle \delta u_L^3 \rangle = \frac{1}{8} Q , ,
$$
where $Q$ is the downscale enstrophy flux. The important difference between
@eq:K41 and @eq:2DF is that the third order structure function exhibits a
negative linear dependence on $r$ in a downscale energy cascade range and a
positive linear dependence on $r$ in an upscale energy cascade range. Thus, as
argued by @Lindborg1999, the third order structure function could be used in
order to determine the direction of the energy cascade.
@Lindborg1999 calculated second order longitudinal and transverse
structure functions using the MOZAIC dataset. It was
found that the second order structure functions were consistent with
the spectra measured by @NastromGage1985. In a subsequent study,
@ChoLindborg2001 calculated third order structure functions. The
analysis showed reasonably clean results in the stratosphere. It was
found that the third order structure function generally shows a
negative linear dependence on $r$ in the range $r \in [10 , ; 200]$
km, and a positive cubic dependence in the range
$r \in [500, ; 1400]$ km. @ChoLindborg2001 interpreted the negative
linear dependence as a sign of a downscale energy cascade and the
positive cubic dependence as a sign a downscale enstrophy cascade.
The results from the upper troposphere were not as clean as the
results from the lower stratosphere. However, they also indicated
that there is generally a downscale energy cascade at atmospheric
mesoscales. These observations are perhaps the strongest evidence
for the hypothesis that the mesoscale $k^{-5/3}$ spectrum is
associated with a downscale energy cascade.
Stratified turbulence{#sec:strat}
The term stratified turbulence was coined by @Lilly:1983 for flows affected by
a stable, vertical gradient in density or temperature, resulting in
quasi-horizontal motions consisting of large eddies
and gravity waves.
Lilly used the Froude number, $F_v = u/Nl_v$, as an inverse measure of the
strength of the stratification. Here, $u$ is a characteristic horizontal
velocity scale, $N$ is the Brunt-Väisälä frequency and $l_v$ is a vertical
length scale. In particular he considered the limit $F_v \rightarrow 0$.
When a fluid is under geostrophic balance
(equalizing hydrostatic and Coriolis force), one would expect the horizontal
vortices to follow the Taylor-Proudman theorem
[@taylor_motion_1917;@proudman_motion_1916] which is expressed as
$\partial_z \mathbf{u} = 0$. Yet turbulence in strongly stratified fluids have
been observed to evolve into elongated "pancake"-like horizontal layers over
time which decouple the horizontal motion, resulting in vertical variability.
Such structures are thought to be important elements, distinguishing the
physics of stratified turbulence from two-dimensional turbulence
[@riley_fluid_2000]. Understanding the vertical structure of stratified
turbulence is of paramount importance to analyse the energy transport.
![Layers of the atmosphere and typical temperature lapse with altitude (source:
UCAR)](imgs/atmosphere_layers.jpg){#fig:atmo_layer width=75%}
{#fig:atmo_tropo width=75%}
{#fig:atmo_strat width=75%}
Stratification in the atmosphere
The analysis of @riley_direct_1981 and @Lilly:1983, involved a scale decomposition
into waves and vortices with the latter being associated with turbulence.
However, it was unclear what the characteristic vertical length scale of
stratified turbulence is. Experimental
[@fincham_energy_1996;@billant_experimental_2000] and numerical [@herring_numerical_1989]
investigations of stratified turbulence demonstrated that thin layered
structures can emerge from purely horizontal base flows or forcing.
Strongly stratified flows with sufficiently high Reynolds number show
a "zig-zag instability" with alternating elongated horizontal structures.
The vertical length scale of these was found to scale as $u/N$.
In other words, the vertical Froude number used by Lilly as an inverse measure
of the strength of the stratification, would typically be of the order of
unity. @Billant2001 instead introduced a Froude number based on a horizontal
length scale, $F_h = u/Nl_h$, as an inverse measure of the strength of the
stratification. Introducing an advective time scale $T = l_h/u$ and a verical
length scale $l_v \ge u/N$, the authors
demonstrate that it is possible to simplify the Boussinesq equations into a set
of dimensionless equations describing stratified turbulence. Furthermore,
assuming that the aspect ratio parameter, $\delta = l_v/l_h$, scales as
$\delta \sim F_h$ or $l_v = u/N$ in the limit of strong stratification, $F_h
\rightarrow 0$, a set of self-similar reduced-order equations were also
derived. These equations are invariant under a group of
transformations of $N, z, u_z$ and $\rho$. The implications of this seminal
study was that, unlike the explanation in @Lilly:1983, by introducing a vertical
length scale, $l_v = u / N$, one can derive a reduced-order system in which
waves and vortices evolve on similar time and length scales. @Billant2001 also
suggested that the buoyancy Reynolds number $Re_b = Re_h F_h^2$, where $Re_h$
is the Reynolds number based on a horizontal length scale, must meet the
condition $1 \ll Re_b \ll Re_h$, for the self-similarity to hold. These
results were also reproduced
in decaying [@riley_dynamics_2003] and forced
[@Waite-Bartello:2004;@Lindborg2006] simulations of stratified turbulence. In
@Lindborg2006, it was also shown that the energy cascade is similar to
Kolmogorov's turbulence picture, and that the kinetic and potential energy
spectra scale as:
$$
E_K(k_h) = C_K \epsilon_K^{2/3} k_h^{-5/3},
$${#eq:EKspectra}
$$
E_P(k_h) = C_P \epsilon_K^{-1/3} \epsilon_P k_h^{-5/3},
$${#eq:EPspectra}
where, $\epsilon_K$ and $\epsilon_P$ are rates of dissipation of kinetic and
potential energies respectively and the coefficients were found from
simulations to be $C_K \approx C_P \approx 0.5$. Approximate equipartition,
$E_K \sim E_P$ and $\epsilon_K \sim \epsilon_P$, was assumed and total
dissipation was expected to scale as, $\epsilon = \epsilon_K + \epsilon_P \sim
u^3 / l_h$ [@taylor_statistical_1935]. For flow structures smaller than the
Ozmidov length scale, $l_0 = \epsilon^{1/2} / N^{3/2}$, stratified turbulence
transitions to local patches of isotropic turbulence.
@Lindborg2006 argued that the stratified turbulence theory can explain the mesoscale
spectra reported in @NastromGage1985. The spectral energy fluxes were positive
for all wavenumbers in the simulations
implying a forward energy cascade as @Dewan:1979 initially anticipated.
However, with $E_K(k_h) \sim 3 E_P(k_h)$ there was no perfect equipartition of
energy between kinetic and potential energy
suggesting that the physical mechanism is not linear gravity waves but
nonlinear layered motions resulting from stratified turbulence. In the light
of these new results, one can say that the majority of results tend to favour the
forward cascade hypothesis. Stratified turbulence might be a possible
explanation of mesoscale spectra, but as observed from the GCM simulations of
@AugierLindborg2013, it might not be necessary to simulate a fully resolved
stratified flow to mimic its dynamics. We shall now see how a simpler model is
used in the present thesis to study geophysical turbulence.
Footnotes
\fullcite{NastromGage1985}. \textcopyright American
Meteorological Society. Used with permission. ↩