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%                    B O F / N U M E R I C . T E X 
%                    doc: Fri Apr  7 21:13:27 2000
%                    dlm: Thu Jun  8 00:24:44 2000
%                    (c) 2000 A.M. Thurnherr
%                    uE-Info: 66 12 NIL 0 0 72 3 2 4 ofnI
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\begin{figure}[t]

\placeFig{BOF/sawtooth}

\caption[Model responses to sawtooth forcing]{Model responses to
sawtooth forcing with (non-dimensional) amplitude 1, fluctuating around
a mean of 1; responses to forcing with period 0.5 (not shown) and
$\Bhzero=2\Bheq$, and to forcing with period 1 and $\Bhzero=0$ are
plotted; analytic solutions to steady-state forcing $\BQi-\BQu=1$ are
shown for comparison.}

\labelFig{BOF-sawtooth}

\end{figure}
%--------------------

To investigate the model responses to oscillatory forcing
\rexpr{BOF-model} is solved numerically for a variety of initial
conditions, forcing amplitudes, frequencies and shapes.
\rFigNP{BOF-sawtooth} shows model responses to sawtooth forcing with
(non-dimensional) periods 0.5 and 1, and amplitudes 1 oscillating
around a mean of $\BQi-\BQu=1$. The model basin acts as an effective
low-pass filter as indicated by the close resemblance of the numerical
and the corresponding analytical solutions (for steady forcing with the
same mean and the same initial conditions), by the near-sinusoidal
responses to sawtooth forcing, and by the reduction of the model
response amplitudes with increasing forcing frequency.

%--------------------
\begin{figure}[t]

\placeFig{BOF/events}

\caption[Model responses to event-like forcing]{Model adjustment to
event-like deviations from the steady state $\BQi-\BQu=1$; responses to
positive and negative triangular spikes of amplitudes $\pm0.6$ and
durations 3 are plotted.}

\labelFig{BOF-events}

\end{figure}
%--------------------
\REM{\begin{figure}[t]

\placeFig{BOF/response}

\caption[Model response as a function of spike duration]{Normalized
maximum model response as a function of spike duration for positive and
negative spikes of 0.1 and 1.0 event amplitudes.}

\labelFig{BOF-response}

\end{figure}
%--------------------
ENDREM}

To explore the responses of the model outflow to event-like
disturbances \rexpr{BOF-model} was solved numerically for a variety of
deviations (of different shapes, durations and amplitudes) from the
steady state. \rFigNP{BOF-events} shows two examples. The model outflow
responds stronger to positive than to negative events and there is a
temporal asymmetry in both cases. Responses to different event shapes
are similar and the model response amplitudes decrease with decreasing
event durations, consistent with the low-pass filtering property of the
basin/sill combination.

\REM{To assess the outflow response quantitatively as a function of
event duration and amplitude, the maximum deviations from the steady
state normalized by the event amplitudes are shown for both positive
and negative ``spikes'' in \rfigNP{BOF-response}. The differences are
generally small but somewhat greater in negative events where the
normalized response decreases with increasing amplitude whereas it
increases in the case of positive events.}

