dirichlet_modes.tex

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\title{On the Dirichlet modes of Scott and Furnival JPO 2012}
\author{Cesar B Rocha\thanks{Scripps Institution of Oceanography, University of California, San Diego; \texttt{crocha@ucsd.edu}}}
\date{\today}

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The ``Dirchlet modes'' Scott and Furnival JPO 2012 (hereafter SF12) should not
have zero slope at the surface unless the buoyancy frequency is zero or behaves
wildy at the surface. The authors do not show the buoyancy profile used in
their calculations. In an appendix the authors do mention that the buoyancy
frequency at the surface is very small. This can be account for the seemly
``flat'' slope of the ``Dirichlet modes'' near the surface (see their figure 1b).

\section{WKB approximate solution to ``Dirichlet modes''}

The ``'Dirichlet modes'', here denoted $\sD_n$, satisfy

\beq
\label{dirich_eigpb}
\frac{\dd}{\dd z}\left[\bur \frac{\dd\sD_n}{\dd z} \right] = -\kappa_n^2 \sD_n\com
\eeq
with boundary conditions
\beq
\label{dirich_bc}
@z = -H:\qquad \frac{\dd \sD_n}{\dd z}= 0\com
\eeq
and
\beq
\label{dirich_bc0}
@z = 0:\qquad \sD_n = 0 \per
\eeq
Equation \eqref{dirich_eigpb} can be rewritten as
\beq
\label{dirich_eigpb_wkb}
\bur \frac{\dd^2\sD_n}{\dd z^2} + \frac{\dd \sD_n}{\dd z}\,\frac{\dd}{\dd z}\bur + \kappa_n^2 \sD_n = 0\per
\eeq
We are going to obtain WKB approximate solution to \eqref{dirich_eigpb_wkb}. We use the following notation

\beq
\label{notation}
\ep \defn \frac{1}{\kappa_n} \qquad \text{and} \qquad S^2(z) \defn \ibur \per 
\eeq
With the definitions in \eqref{notation} we have the renotated equation
\beq
\label{dirich_eigpb_wkb_ep}
\ep^2\,\frac{\dd^2\sD_n}{\dd z^2} - \frac{\dd}{\dd z}\left(\log S^2(z)\right)  \frac{\dd \sD_n}{\dd z} + S^2(z) \sD_n = 0\per
\eeq
In the WKB spirit we assume that $S^2(z)$ is slowly varying i.e., the buoyancy frequency $N^2(z)$ does not vary very fast. (This assumption may be problematic near the base of the mixed-layer.) We also assume that $\ep$ is small; the accuracy of the WKB solution improves with mode number. We now make the exponential approximation (e.g., Bender and Orszag)
\beq
\sD_n^e \defn \ee^{Q(z)/\ep}\per
\eeq
Thus we have
\beq
{\sD_n^e}' = \frac{Q'(z)}{\ep}\sD_n^e\com
\eeq
and 
\beq
{\sD_n^e}'' = \left[\left(\frac{Q'(z)}{\ep}\right)^2 + \frac{Q''(z)}{\ep} \right]\sD_n^e\com
\eeq
where primes represent differentiation with respect to $z$.
We now expand $Q(z)$ in powers of $\ep$

\beq
\label{aseries}
Q(z) = Q_0(z)  + \ep\,Q_1(z) + \ep^2\,Q_2(z) + \mathcal{O}(\ep^3)\per
\eeq
Substituting \eqref{aseries} in \eqref{dirich_eigpb_wkb_ep} we obtain, to lowest order $\mathcal{O}(\ep^0)$,
\beq
\label{lowest_order_eqn}
Q_0'^2 + S^2(z) = 0\per
\eeq
Thus
\beq
\label{Q0}
Q_0 = \pm \ii \int^z \!\!\!S(\xi) \,\dd \xi \per
\eeq
At $\mathcal{O}(\ep)$ we have
\beq
\label{first_order_eqn}
2\,Q_0'Q_1' + Q_0'' - Q_0' \frac{\dd}{\dd z} \left(\log S^2(z)\right) = 0\per
\eeq
Hence
\beq
\label{Q_1}
Q_1  =   \frac{1}{2} \left(\log S^2(z)\right) - \frac{1}{2} \left(\log \pm \ii S(z) \right) + \text{const}\,\, \per
\eeq
Notice that the imaginary part in the $\log$ in \eqref{Q_1} just contributes to the irrelevant constant. Thus
\beq
Q_1 = \log \sqrt{S(z)} + \text{const} \,\, \per
\eeq
In the  ``physical optics'' approximation we keep the first two terms in the expansion \eqref{aseries}. The solution to \eqref{dirich_eigpb_wkb_ep}, consistent with the bottom boundary condition \eqref{dirich_bc}, is

\beq
D_n^{po} = A_n\, \sqrt{N(z)}\, \cos \left(\frac{\kappa_n}{f_0} \int_{-H}^{z} \!\!\!N(\xi) \dd \xi\right)\com
\eeq
where $A_n$ is a constant. By imposing the boundary condition at $z=0$ \eqref{dirich_bc0}, we obtain $\kappa_n$:

\beq
\label{kappan}
\kappa_n = \frac{(n + 1/2) \pi \, f_0}{\int_{-H}^0N(\xi)\dd \xi} \com\qquad n=0,1,2,\ldots 
\eeq

The constant $A_n$ is determined by the normalization condition 

\beq
\frac{1}{H}\int_{-H}^{0}\!\!\!\sD_n\,\sD_m \dd z = \delta_{mn}\com 
\eeq
which gives 
\beq
\label{an_eqn}
A_n^2 \, \int_{-H}^{0}\!\! N(z) \cos \left(\frac{\kappa_n}{f_0}\int_{-H}^{z}\!\!\!N(\xi) \dd \xi\right) \dd z = H\per
\eeq
The integral in \eqref{an_eqn} can be evaluated exactly by making the change of variables 

\beq
\eta \defn \cos \left(\frac{\kappa_n}{f_0}\int_{-H}^{z}\!\!\! N(\xi) \dd \xi \right)\com\qquad \dd\eta = \frac{\kappa_n}{f_0}N(z) \dd z\com
\eeq
and using the expression for the eigenvalues \eqref{kappan}. We obtain 
\beq
A_n = \left[\frac{2\,H}{\int_{-H}^{0}\! N(\xi) \dd \xi} \right]^{1/2}
\eeq
Thus the WKB approximate solution to the ``Dirichlet modes'' is
\beq
D_n^{po} = \left[\frac{2\,N(z)\,H}{\int_{-H}^{0}\! N(\xi) \dd \xi} \right]^{1/2}\!\!\cos \left( \frac{(n + 1/2)\pi}{\int_{-H}^{0} \!N(\xi) \dd \xi} \,\,\,\int_{-H}^{z} \!N(\xi) \dd \xi\right)\per
\eeq
The slope of $D_n^{po}$ is, to lowest order,
\beq
\frac{\dd D_n^{po}}{\dd z} =  \frac{\sqrt{2} \, N(z)\,(n+1/2)\pi}{\int_{-H}^0 N(\xi) \dd \xi}  \sin \left(\frac{(n + 1/2)\pi}{\int_{-H}^{0} \!N(\xi) \dd \xi} \,\,\,\int_{-H}^{z} \!N(\xi) \dd \xi\right)\per
\eeq

\subsection*{The constant buoyancy frequency limit}
With $N = \text{const.}$ we obtain

\beq
D_n^{po} = \sqrt{2} \cos\left[\pi\,(n+1/2)(1+z/H)\right]\com
\eeq
and

\beq
\frac{\dd D_n^{po}}{\dd z}\Bigg|_{z=0} = \sqrt{2}\,(n+1/2)\pi \sin \left[\pi\,(n+1/2)(1+z/H)\right]\per
\eeq
The slope at the surface is $\pm\sqrt{2} \,(n+1/2)\pi$ which is clearly non-zero.

\subsection*{Non-constant stratification}
For variable buoyancy profile the slope at the surface is
\beq
\frac{\dd D_n^{po}}{\dd z}\Bigg|_{z=0} =  \frac{\sqrt{2} \,N(0) \, H\,(n+1/2)\pi}{\int_{-H}^0 N(\xi) \dd \xi}\per
\eeq
The slope of the ``Dirichlet modes'' at the surface depends on the surface buoyancy frequency which can be problematic. The slope if $N(z)$ vanishes at the surface. Moreover, if $N(z)$ varies rapidly near the surface, the WKB approximation is not accurate.       

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