Definitions of variables

In SWAN a number of variables are used in input and output. Most of them are related to waves. The definitions of these variables are mostly conventional.

HSIGN	Significant wave height, denoted as $H_s$ in meters, and defined as
	$H_s = 4 \sqrt{\int \int E(\omega, \theta) d\omega d\theta}$
	where $E(\omega,\theta)$ is the variance density spectrum and $\omega$ is the absolute
	radian frequency determined by the Doppler shifted dispersion relation.
	However, for ease of computation, $H_s$ can be determined as follows:
	$H_s = 4 \sqrt{\int \int E(\sigma, \theta) d\sigma d\theta}$
HSWELL	Significant wave height associated with the low frequency part of
	the spectrum, denoted as $H_{s,{\rm swell}}$ in meters, and defined as
	$H_{s,{\rm swell}} = 4 \sqrt{\int_{0}^{\omega_{\rm swell}} \int_{0}^{2\pi} E(\omega, \theta) d\omega d\theta}$
	with $\omega_{\rm swell}=2\pi f_{\rm swell}$ and $f_{\rm swell}=0.1$ Hz by default (this can be changed
	with the command QUANTITY).
TMM10	Mean absolute wave period (in s) of $E(\omega,\theta)$, defined as
	$T_{m-10} = 2\pi \frac{\int \int \omega^{-1} E(\omega, \theta) d\omega d\theta}{... ...E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta}$
TM01	Mean absolute wave period (in s) of $E(\omega,\theta)$, defined as
	$T_{m01} = 2\pi \left(\frac{\int \int \omega E(\omega, \theta) d\omega d\theta}{... ...eta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} \right)^{-1}$
TM02	Mean absolute wave period (in s) of $E(\omega,\theta)$, defined as
	$T_{m02} = 2\pi \left(\frac{\int \int \omega^2 E(\omega, \theta) d\omega d\theta... ...a) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} \right)^{-1/2}$
DIR	Mean wave direction (in $^o$, Cartesian or Nautical convention),
	as defined by (see Kuik et al. (1988)):
	${\rm DIR} = \frac{180}{\pi} \, \arctan \left ( \frac{\int \sin \theta E(\sigma,... ...) d\sigma d\theta}{\int \cos \theta E(\sigma, \theta) d\sigma d\theta} \right )$
	This direction is the direction normal to the wave crests.
PDIR	Peak direction of $E(\theta) = \int E(\omega,\theta)d\omega = \int E(\sigma,\theta)d\sigma$
	(in $^o$, Cartesian or Nautical convention).
TDIR	Direction of energy transport (in $^o$, Cartesian or Nautical convention).
	Note that if currents are present, TDIR is different from the mean wave
	direction DIR.
RTMM10	Mean relative wave period (in s) of $E(\sigma,\theta)$, defined as
	$RT_{m-10} = 2\pi \frac{\int \int \sigma^{-1} E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta}$
	This is equal to TMM10 in the absence of currents.
RTM01	Mean relative wave period (in s) of $E(\sigma,\theta)$, defined as
	$RT_{m01} = 2\pi \left(\frac{\int \int \sigma E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} \right)^{-1}$
	This is equal to TM01 in the absence of currents.
RTP	Relative peak period (in s) of $E(\sigma)$ (equal to absolute peak period
	in the absence of currents).
	Note that this peak period is related to the absolute maximum bin of the
	discrete wave spectrum and hence, might not be the 'real' peak period.
TPS	Relative peak period (in s) of $E(\sigma)$.
	This value is obtained as the maximum of a parabolic fitting through the
	highest bin and two bins on either side the highest one of the discrete
	wave spectrum. This 'non-discrete' or 'smoothed' value is a better
	estimate of the 'real' peak period compared to the quantity RTP.
PER	Average absolute period (in s) of $E(\omega,\theta)$, defined as
	$T_{m,p-1,p} = 2\pi \frac{\int \int \omega^{p-1} E(\omega, \theta) d\omega d\theta}{\int \int \omega^p E(\omega, \theta) d\omega d\theta}$
	The power $p$ can be chosen by the user by means of the QUANTITY
	command. If $p=1$ (the default value) PER is identical to TM01 and
	if $p=0$, PER = TMM10.
RPER	Average relative period (in s), defined as
	$RT_{m,p-1,p} = 2\pi \frac{\int \int \sigma^{p-1} E(\sigma, \theta) d\sigma d\theta}{\int \int \sigma^p E(\sigma, \theta) d\sigma d\theta}$
	Here, if $p=1$, RPER=RTM01 and if $p=0$, RPER=RTMM10.
FSPR	The normalized frequency width of the spectrum (frequency spreading),
	as defined by Battjes and Van Vledder (1984):
	${\rm FSPR} = \frac{\vert\int_{0}^{\infty} E(\omega) e^{i\omega \tau}d\omega\vert}{E_{\rm tot}} \, , \quad \, \, {\rm for} \, \, \tau = T_{m02}$
DSPR	The one-sided directional width of the spectrum (directional spreading
	or directional standard deviation,in $^o$), defined as
	${\rm DSPR}^2 = \left( \frac{180}{\pi} \right)^2 \int_{0}^{2\pi} (2 \sin (\frac{\theta-\overline{\theta}}{2}))^2 D(\theta) d\theta$
	and computed as conventionally for pitch-and-roll buoy data
	(Kuik et al. (1988); this is the standard definition for WAVEC buoys
	integrated over all frequencies):
	$({\rm DSPR} \frac{\pi}{180})^2 = 2\left( 1 - \sqrt{\left[ \left( \frac{\int\sin... ...\sigma d\theta}{\int E(\sigma,\theta)d\sigma d\theta} \right)^2 \right]}\right)$
QP	The peakedness of the wave spectrum, defined as
	$Q_p = 2 \frac{\int \int \sigma E^2(\sigma, \theta) d\sigma d\theta}{(\int \int E(\sigma, \theta) d\sigma d\theta)^2}$
	This quantity represents the degree of randomness of the waves.
	A smaller value of $Q_p$ indicates a wider spectrum and thus
	increased the degree of randomness (e.g., shorter wave groups),
	whereas a larger value indicates a narrower spectrum and a more
	organised wave field (e.g., longer wave groups).
MS	As input to SWAN with the commands BOUNDPAR and BOUNDSPEC,
	the directional distribution	of incident wave energy is given by
	$D(\theta) = A(\cos\theta)^m$	for all frequencies. The parameter $m$
	is indicated as MS in SWAN and is not necessarily an integer number.
	This number is related to the	one-sided directional spread of the waves
	(DSPR) as follows:

Table A.1: Directional distribution.

MS	DSPR (in $^o$)
1.	37.5
2.	31.5
3.	27.6
4.	24.9
5.	22.9
6.	21.2
7.	19.9
8.	18.8
9.	17.9
10.	17.1
15.	14.2
20.	12.4
30.	10.2
40.	8.9
50.	8.0
60.	7.3
70.	6.8
80.	6.4
90.	6.0
100.	5.7
200.	4.0
400.	2.9
800.	2.0

PROPAGAT	Energy propagation per unit time in $\vec{x}-$, $\theta-$ and $\sigma-$space
	(in W/${\rm m}^2$ or ${\rm m}^2$/s, depending on the command SET).
GENERAT	Energy generation per unit time due to the wind input
	(in W/${\rm m}^2$ or ${\rm m}^2$/s, depending on the command SET).
REDIST	Energy redistribution per unit time due to the sum of quadruplets
	and triads (in W/${\rm m}^2$ or ${\rm m}^2$/s, depending on the command SET).
DISSIP	Energy dissipation per unit time due to the sum of bottom friction,
	whitecapping and depth-induced	surf breaking (in W/${\rm m}^2$ or ${\rm m}^2$/s,
	depending on the command SET).
RADSTR	Work done by the radiation stress per unit time, defined as
	$\int\limits_{0}\limits^{2\pi} \int\limits_{\sigma_{\mbox{\tiny low}}}\limits^{\... ...la_{(\sigma,\theta)} \cdot ({\vec{c}}_{(\sigma,\theta)} E)\vert d\sigma d\theta$
	(in W/${\rm m}^2$ or ${\rm m}^2$/s, depending on the command SET).
WLEN	The mean wave length, defined as
	${\rm WLEN} = 2\pi \left( \frac{\int \int k^{p} E(\sigma,\theta)d\sigma d\theta}{\int \int k^{p-1} E(\sigma,\theta)d\sigma d\theta} \right)^{-1}$
	As default, $p=1$ (see command QUANTITY).
STEEPNESS	Wave steepness computed as HSIG/WLEN.
BFI	The Benjamin-Feir index or the steepness-over-randomness ratio,
	defined as
	${\rm BFI} = \sqrt{2\pi} \times$ STEEPNESS $\times$ QP
	This index can be used to quantify the probability of freak waves.
QB	Fraction of breakers in expression of Battjes and Janssen (1978).
TRANSP	Energy transport with components $P_x = \rho g\int \int c_x E(\sigma,\theta)d\sigma d\theta$ and
	$P_y = \rho g\int \int c_y E(\sigma,\theta)d\sigma d\theta$	with $x$ and $y$ the problem coordinate system,
	except in the case of output with BLOCK command in combination
	with command FRAME, where $x$ and $y$ relate to the $x-$axis and $y-$axis
	of the output frame.
VEL	Current velocity components in $x-$ and $y-$direction of the problem
	coordinate system,	except in the case of output with BLOCK command in
	combination with command FRAME, where $x$ and $y$ relate to the $x-$axis
	and $y-$axis of the output frame.
WIND	Wind velocity components in $x-$ and $y-$direction of the problem coordinate
	sytem, except in the case of output with BLOCK command in	combination
	with command FRAME, where $x$ and $y$ relate to the $x-$axis and $y-$axis of
	the output frame.
FORCE	Wave-induced force per unit surface area (gradient of radiation stresses)
	with $x$ and $y$ the problem coordinate system, except in the case of output
	with BLOCK command in	combination with command FRAME,
	where $x$ and $y$ relate to the $x-$axis and $y-$axis of the output frame.
	$F_x = -\frac{\partial S_{xx}}{\partial x} - \frac{\partial S_{xy}}{\partial y}$
	$F_y = -\frac{\partial S_{yx}}{\partial x} - \frac{\partial S_{yy}}{\partial y}$
	where $S$ is the radiation stress tensor as given by
	$S_{xx} = \rho g \int ( n \cos^2\theta + n - \frac{1}{2} ) E d\sigma d\theta$
	$S_{xy} = S_{yx} = \rho g \int n \sin\theta \cos\theta E d\sigma d\theta$
	$S_{yy} = \rho g \int ( n \sin^2\theta + n - \frac{1}{2} ) E d\sigma d\theta$
	and $n$ is the group velocity over the phase velocity.
UBOT	Root-mean-square value (in m/s) of the maxima of the orbital motion
	near the bottom $U_{\rm bot} = \sqrt{2} U_{\rm rms}$.
URMS	Root-mean-square value (in m/s) of the orbital motion near the bottom.
	$U_{\rm rms} = \sqrt{\int_{0}^{2\pi} \int_{0}^{\infty} \frac{\sigma^2}{\sinh^2 kd} E(\sigma,\theta) d\sigma d\theta}$
TMBOT	Near bottom wave period (in s) defined as the ratio of the bottom excursion
	amplitude to the root-mean-square velocity $T_{\rm b} = \sqrt{2}\pi a_{\rm b}/U_{\rm rms}$ with
	$a_{\rm b} = \sqrt{2\int_{0}^{2\pi} \int_{0}^{\infty} \frac{1}{\sinh^2 kd} E(\sigma,\theta) d\sigma d\theta}$
LEAK	Numerical loss of energy equal to $c_{\theta} E(\omega,\theta)$ across boundaries $\theta_1$=[dir1]
	and $\theta_2$=[dir2] of a directional sector (see command CGRID).
TIME	Full date-time string.
TSEC	Time in seconds with respect to a reference time (see command QUANTITY).
SETUP	The elevation of mean water level (relative to still water level) induced by
	the gradient of the radiation stresses of the waves.
Cartesian convention	The direction is the angle between the vector and the positive $x-$axis,
	measured counterclockwise. In other words: the direction where the
	waves are going to or where the wind is blowing to.
Nautical convention	The direction of the vector from geographic North measured
	clockwise. In other words: the direction where the waves are coming
	from or where the wind is blowing from.