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Atmospheric turbulence: Shallow Water and Toy Model Equations
2D turbulence
It stated that a Gaussian initial state with an energy spectra $E(k) = \pi k
U(k)$ would reach an equilibrium distribution which scales as $U(k)~k^{n}$
where $n \notin [0, 2]$. Therefore, it implies a bidirectional transfer of
energy starting from the initial intermediate scale.
Of particular interest was also the note that while the \kfivethird\ spectrum
could arise from local interactions, the $k^{-3}$ spectrum would be non-local in
nature. Kraichnan identified that these results would have deep
impact in our understanding of mesoscale turbulence.
Quasi-geostrophic turbulence
Rossby number $Ro < \order{1}$, indicating strong rotation and the flow in
near-geostrophic balance,
Combining this with a scaling analysis of the hydrostatic
relation for a Boussinesq fluid, which is an equillibrium between pressure
gradient and Coriolis force,
\begin{align*}
| \nabla \phi' | &\sim | {\bf f} \times {\bf u} |\
\phi' &\sim f_o U L \
b' = \p_z{\phi'} &\sim \frac{f_0 U L}{H}\
\frac{\p_zb'}{N^2} &\sim \frac{f_0 U L}{H^2N^2}
\sim Ro \left( \frac{f_0L}{HN} \right)^2
\sim Ro \left( \frac{L}{L_d} \right)^2 \
\therefore \frac{\p_zb'}{N^2} &\sim \order{Ro} < \order{1}
\end{align*}
where $b'$ stands for fluctuations in buoyancy and its evolution equation is
$\Dt b' = - N^2 w$.
This scaling would imply that variations in
stratification are small in comparison with background stratification.
Atmospheric energetics
@NastromGage1985 which compiled data from over 6000 aircraft flights spanning several years,
@Lindborg1999 argues that, it is still possible to accommodate both $k^{-3}$
and \kfivethird\ ranges without adding a sink at intermediate scales. He
demonstrates the possibility of a inertial range determined by both
constant-energy and enstrophy fluxes, by allowing both a large scale and a
small energy source to exists.
An important contribution introduced in @Lindborg1999 was analytical relations
for second-, third- and fourth-order structure functions for two-dimensional
turbulence. Structure functions were used instead of spectral analysis, as it
can be applied on one-dimensional non-uniform data and does not require removal
of the mean flow. These relations were then tested upon MOZAIC
(Measurement of Ozone by Airbus in-service aircraft) dataset to investigate
whether the power law scaling in the inertial range, the direction of the
cascade, and the intermittency can be explained within the framework of
two-dimensional turbulence theory or not. The measurements indicate an
agreement with second-order structure relation,
$$\braket{\delta {\bf u} \cdot \delta {\bf u}}(r) = C_u \epsilon_K^{2/3}
r^{2/3}$${#eq:structfn2}
where, $\epsilon_K$ is the kinetic energy flux. The $r^{2/3}$ scaling is
equivalent to a $k^{-5/3}$ spectrum. @Lindborg1999 also remarked that the
scaling was found to fit better for larger separations and was conjectured to
be caused by three-dimensional effects becoming influential at smaller scales.
The third order structure function relation,
$$\braket{(\delta u_L)^3} + \braket{\delta u_L(\delta u_T)^2}(r) = 2P_S + \frac{1}{4}Q_Lr^3$${#eq:structfn3}
where $P_S$ is a small scale forcing of kinetic energy and $Q_L$ is a large
scale forcing of enstrophy, is equivalent to spectral energy fluxes and
positive values of right hand side imply inverse energy cascade and forward
enstrophy cascade. These third order structure functions reported in
@Lindborg1999 were later on, correctly computed in @ChoLindborg2001 as a test
for both the direction of cascade and two-dimensional turbulence. The analysis
separated the data into troposphere and stratosphere and also into five
latitudinal (or zonal) bands to account for inhomogeneities.
The analysis strongly suggests that in the stratosphere, for the range $10 < r
< 150$ km the third order structure function is generally negative and scales
as $-r$, implying a forward energy cascade; and in the range $540 < r < 1400$
km the same scales as $r^3$ with positive values, implying a forward enstrophy
flux.
The authors also indicate that the overall tropospheric structure function were
negative for all mesoscales, although it did not converge to follow a
particular power law. These observations are perhaps the strongest evidence for
the forward energy cascade hypothesis for the mesoscales.
Next, we turn to question of what resolution is required to reproduce the
mesoscale spectrum and associated physics.
Stratified turbulence
The troposphere, the lowest layer of the atmosphere, is relatively thin (10 to 20 km)
compared to Earth's radius 6400 km. Fluid motion is predominantly horizontal.
The troposphere is also constantly influenced by background stratification resulting
from the vertical potential temperature gradient. Stratified turbulence is a discipline
of fluid mechanics which studies such flows.
Stratification is characterized by the
Brunt-\text{V"ais"al"a} frequency, $N=\sqrt{\frac{-g}{\rho_0}
\pder[\rho(z)]{z}}$ and the Froude number based on horizontal velocity and
length scale, $F_h = u / (Nl_h)$.
vertical length scale must depend on characteristic properties of the flow.
Zig-zag instability led @Billant2001 to propose a new scaling for stratified turbulence,
wherein, the vertical length scale of the turbulence,
leads to, was postulated to scale as, $l_v = u / N$. Using this result, along
with two hypotheses: $F_h \ll 1$ and advective time scale based on horizontal
length scale $T = l_h / u$, the authors demonstrate that it is possible to
simplify the Boussinesq equations into a set of dimensionless equations
describing stratified turbulence. Furthermore, enforcing
the aspect ratio parameter $\delta = l_v / l_h = F_h$ at the limit of strong
stratification $Fr \to 0$, a set of self-similar
SWE
Of the two positive-definite terms in the above expression, the first term
$c^2/2$ is invariant.
The last term $E_A = c^2\eta^2/2$ is a measure of the
potential energy due to surface displacement and this would through fluid
motion be converted back to kinetic energy. This term is called available
potential energy (APE) [@Lorenz:1955].
This would naturally mean that higher powers of $Q$ would be conserved,
including the quadratic $Q^2$ potential enstrophy.
Following Farge & Sadourny 1989
Instead of finding the normal modes for the vorticity, divergence and
displacement field of the flow, we shall make use of the Helmholtz
decomposition described in [@eq:helm_u]. The shallow water equations
then transform to:
\begin{align}
\partial_t \psi = & f \phi
\label{eq:dtpsi_l} \
\partial_t \phi = & -f \psi - c^2 \eta \label{eq:dtphi_l} \
\partial_t \eta = & - \nabla^2 \phi \label{eq:dteta_l2}\end{align}
where $\psi$ and $\psi$ are stream function and velocity potential as
functions of $\mathbf{r}$ and $t$ respectively. By substituting the
dependent variables with the respective Fourier transform, this reduces
to the eigenvalue problem:
\begin{align*}
i\omega
\begin{Bmatrix}
\hat{\psi} \ \hat{\phi} \ \hat{\eta}
\end{Bmatrix}
= i
\begin{bmatrix}
0 & if & 0 \
-if & 0 & ic^2 \
0 & -i\kappa^2 & 0
\end{bmatrix}
\begin{Bmatrix}
\hat{\psi} \ \hat{\phi} \ \hat{\eta}
\end{Bmatrix}\end{align*}
the square matrix is not Hermitian and this would result in complex
eigenvalues. By adopting the following change of variables:
$$\hat{\psi} \to \kappa^2\hat \psi = \hat{\zeta} ; \quad
\hat{\phi} \to -\kappa^2\hat \phi = \hat{\delta}; \quad
\hat{\eta} \to c\kappa\hat \eta$$ it falls back to the previous
eigenvalue problem as demonstrated in the previous section. In other
words, we can use the same eigenvector matrix, $X_n$ to find the normal
modes of:
$$\mathbf{H} = {\hat \psi,; \hat \phi,; \eta }^T$$ which is
closely related to: $$\mathbf{W}
= {\kappa^2\hat \psi,; -\kappa^2\hat
\phi,; c\kappa\hat \eta }^T
= {\hat \zeta;; \hat \delta;; c\kappa \hat \eta }^T$$
Toy model
Replace the right hand side of the scalar equation to make it linear.
Reason: $\eta << 1$ when $Fr -> 0$
Use Helmholtz decomposition to calculate
$\bf{u}^r$, ${\bf u} = \bf{u}^r + \bf{u}^d$ i.e. in the advection term we
use rotational velocity. Reason: large scale motions dominated by rotation.
Spectral energy budget
Spectral energy budget is a statistical analysis of the direction of energy
cascade, or in other words energy flux as a function of wavenumber and also
conversion of energy between different modes.
While using pseudospectral methods is advantageous to to solve the shallow
water equations in normal-modes, for a more accurate representation of the
viscous term1 and faster computation. Therefore note that the we have
Results
In both cases, the forcing is narrow band in space,
around a wavenumber $k_f$, and random in time. This is indicated by the steep
positive jump at $k/k_f = 1$. The run on the left is primarily forced in the
wave modes, in particular, using the ageostrophic variable $a = f \zeta - c
\nabla^2 \theta$. As a result the strong nonlinear wave energy cascade as shown
by dominance of $\Pi_{VWW}$ in almost all scales and $\Pi_{WWW}$ in
intermediate and small scales. On the right, we force in available potential
energy $E_P$, and due to system rotation, the potential-vortical modes are
also excited. As a consequence we see that the large scales are dominated by
$\Pi_{VVW}$ and smaller scales by $\Pi_{VWW}$. The total energy flux $Pi$ is
positive at scales smaller than $k_f$, which implies forward energy cascade.