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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 Hs in meters, and defined as  
     
  Hs = 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, Hs can be determined as follows:  
     
  Hs = 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 Hs,swell in meters, and defined as  
     
  Hs,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$fswell and fswell = 0.1 Hz by default (this can be changed  
  with the command QUANTITY).  
TMM10 Mean absolute wave period (in s) of E($ \omega$,$ \theta$), defined as  
     
  Tm-10 = 2$ \pi$$ {\frac{{\int \int \omega^{-1} E(\omega, \theta) d\omega d\theta}}{{\int \int E(\omega, \theta) d\omega d\theta}}}$ = 2$ \pi$$ {\frac{{\int \int \omega^{-1} 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  
     
  Tm01 = 2$ \pi$$ \left(\vphantom{\frac{\int \int \omega E(\omega, \theta) d\omega d\theta}{\int \int E(\omega, \theta) d\omega d\theta} }\right.$$ {\frac{{\int \int \omega E(\omega, \theta) d\omega d\theta}}{{\int \int E(\omega, \theta) d\omega d\theta}}}$$ \left.\vphantom{\frac{\int \int \omega E(\omega, \theta) d\omega d\theta}{\int \int E(\omega, \theta) d\omega d\theta} }\right)^{{-1}}_{}$ = 2$ \pi$$ \left(\vphantom{\frac{\int \int \omega E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} }\right.$$ {\frac{{\int \int \omega E(\sigma, \theta) d\sigma d\theta}}{{\int \int E(\sigma, \theta) d\sigma d\theta}}}$$ \left.\vphantom{\frac{\int \int \omega E(\sigma, \theta) 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  
     
  Tm02 = 2$ \pi$$ \left(\vphantom{\frac{\int \int \omega^2 E(\omega, \theta) d\omega d\theta}{\int \int E(\omega, \theta) d\omega d\theta} }\right.$$ {\frac{{\int \int \omega^2 E(\omega, \theta) d\omega d\theta}}{{\int \int E(\omega, \theta) d\omega d\theta}}}$$ \left.\vphantom{\frac{\int \int \omega^2 E(\omega, \theta) d\omega d\theta}{\int \int E(\omega, \theta) d\omega d\theta} }\right)^{{-1/2}}_{}$ = 2$ \pi$$ \left(\vphantom{\frac{\int \int \omega^2 E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} }\right.$$ {\frac{{\int \int \omega^2 E(\sigma, \theta) d\sigma d\theta}}{{\int \int E(\sigma, \theta) d\sigma d\theta}}}$$ \left.\vphantom{\frac{\int \int \omega^2 E(\sigma, \theta) 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)):  
     
  DIR = $ {\frac{{180}}{{\pi}}}$ arctan$ \left\lfloor\vphantom{ \frac{\int \sin \theta E(\sigma, \theta) d\sigma d\theta}{\int \cos \theta E(\sigma, \theta) d\sigma d\theta} }\right.$$ {\frac{{\int \sin \theta E(\sigma, \theta) d\sigma d\theta}}{{\int \cos \theta E(\sigma, \theta) d\sigma d\theta}}}$$ \left.\vphantom{ \frac{\int \sin \theta E(\sigma, \theta) d\sigma d\theta}{\int \cos \theta E(\sigma, \theta) d\sigma d\theta} }\right\rfloor$  
     
  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  
     
  RTm-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  
     
  RTm01 = 2$ \pi$$ \left(\vphantom{\frac{\int \int \sigma E(\sigma, \theta) d\sigma d\theta}{\int \int E(\sigma, \theta) d\sigma d\theta} }\right.$$ {\frac{{\int \int \sigma E(\sigma, \theta) d\sigma d\theta}}{{\int \int E(\sigma, \theta) d\sigma d\theta}}}$$ \left.\vphantom{\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  
     
  Tm, 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  
     
  RTm, 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):  
     
  FSPR = $ {\frac{{\vert\int_{0}^{\infty} E(\omega) e^{i\omega \tau}d\omega\vert}}{{E_{\rm tot}}}}$ ,      for  $ \tau$ = Tm02  
     
DSPR The one-sided directional width of the spectrum (directional spreading  
  or directional standard deviation,in o), defined as  
     
  DSPR2 = $ \left(\vphantom{ \frac{180}{\pi} }\right.$$ {\frac{{180}}{{\pi}}}$$ \left.\vphantom{ \frac{180}{\pi} }\right)^{2}_{}$$ \int_{{0}}^{{2\pi}}$(2 sin($ {\frac{{\theta-\overline{\theta}}}{{2}}}$))2D($ \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):  
     
  (DSPR$ {\frac{{\pi}}{{180}}}$)2 = 2$ \left(\vphantom{ 1 - \sqrt{\left[ \left( \frac{\int\sin\theta E(\sigma,\theta)...
...sigma d\theta}{\int E(\sigma,\theta)d\sigma d\theta} \right)^2 \right]}}\right.$1 - $ \sqrt{{\left[ \left( \frac{\int\sin\theta E(\sigma,\theta)d\sigma d\theta}{\in...
...heta)d\sigma d\theta}{\int E(\sigma,\theta)d\sigma d\theta} \right)^2 \right]}}$$ \left.\vphantom{ 1 - \sqrt{\left[ \left( \frac{\int\sin\theta E(\sigma,\theta)...
...sigma d\theta}{\int E(\sigma,\theta)d\sigma d\theta} \right)^2 \right]}}\right)$  
     
QP The peakedness of the wave spectrum, defined as  
     
  Qp = 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 Qp 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/m2 or m2/s, depending on the command SET).  
GENERAT Energy generation per unit time due to the wind input  
  (in W/m2 or m2/s, depending on the command SET).  
REDIST Energy redistribution per unit time due to the sum of quadruplets  
  and triads (in W/m2 or m2/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/m2 or m2/s,
  depending on the command SET).  
RADSTR Work done by the radiation stress per unit time, defined as  
     
  $ \int\limits_{{0}}^{}$2$\scriptstyle \pi$$ \int\limits_{{\sigma_{\mbox{\tiny low}}}}^{}$$\scriptstyle \sigma_{{\mbox{\tiny high}}}$| Stot - $ {\frac{{\partial E}}{{\partial t}}}$ - $ \nabla_{{\vec{x}}}^{}$ . [($ \vec{{c}}_{{g}}^{{}}$ + $ \vec{{U}}\,$)E] - $ \nabla_{{(\sigma,\theta)}}^{}$ . ($ \vec{{c}}_{{{(\sigma,\theta)}}}^{{}}$E)| d$ \sigma$d$ \theta$  
     
  (in W/m2 or m2/s, depending on the command SET).  
WLEN The mean wavelength, defined as  
     
  WLEN = 2$ \pi$$ \left(\vphantom{ \frac{\int \int k^{p} E(\sigma,\theta)d\sigma d\theta}{\int \int k^{p-1} E(\sigma,\theta)d\sigma d\theta} }\right.$$ {\frac{{\int \int k^{p} E(\sigma,\theta)d\sigma d\theta}}{{\int \int k^{p-1} E(\sigma,\theta)d\sigma d\theta}}}$$ \left.\vphantom{ \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  
     
  BFI = $ \sqrt{{2\pi}}$ x STEEPNESS x 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 Px = $ \rho$g$ \int$$ \int$cxE($ \sigma$,$ \theta$)d$ \sigma$d$ \theta$ and  
  Py = $ \rho$g$ \int$$ \int$cyE($ \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.  
     
  Fx = - $ {\frac{{\partial S_{xx}}}{{\partial x}}}$ - $ {\frac{{\partial S_{xy}}}{{\partial y}}}$  
     
  Fy = - $ {\frac{{\partial S_{yx}}}{{\partial x}}}$ - $ {\frac{{\partial S_{yy}}}{{\partial y}}}$  
     
  where S is the radiation stress tensor as given by  
     
  Sxx = $ \rho$g$ \int$$ \lfloor$n cos2$ \theta$ + n - $ {\frac{{1}}{{2}}}$$ \rfloor$Ed$ \sigma$d$ \theta$  
     
  Sxy = Syx = $ \rho$g$ \int$n sin$ \theta$cos$ \theta$Ed$ \sigma$d$ \theta$  
     
  Syy = $ \rho$g$ \int$$ \lfloor$n sin2$ \theta$ + n - $ {\frac{{1}}{{2}}}$$ \rfloor$Ed$ \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 Ubot = $ \sqrt{{2}}$Urms.  
URMS Root-mean-square value (in m/s) of the orbital motion near the bottom.  
     
  Urms = $ \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 Tb = $ \sqrt{{2}}$$ \pi$ab/Urms with  
     
  ab = $ \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$\scriptstyle \theta$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.  


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The SWAN team 2017-10-26