chapter_00_0_intro.md

\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.

{#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

1. \fullcite{NastromGage1985}. \textcopyright American Meteorological Society. Used with permission. ↩