The Fully Developed Wind-Sea Spectrum as a Solution of the Energy Balance Equation View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1985

AUTHORS

G. J. Kamen , S. Hasselmann , K. Hasselmann

ABSTRACT

We consider the energy transfer equation for well developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parametrization proposed by Snyder et al (1981), which was based on their measurements in the Bight of Abaco, and which agrees with Miles’ (1957, 1959) theory. The wave-wave interaction is computed with an algorithm given by Hasselmann et al (1984). The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters (cf. Hasselmann, 1974). Full details of this study are given elsewhere (Kamen, Hasselmann and Hasselmann, 1984). Here we summarize the main results. In the first part of our study we investigated whether the assumption that the equilibrium spectrum exists and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. This is illustrated in fig. 1, in which the assumed asymptotic spectrum and the corresponding source terms are given. The dissipation constant is chosen such that at any given frequency the total rate of change vanishes. As one can see there is no angular balance. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the study we chose a different approach. We assumed that the dissipation was given and we performed numerical experiments simulating fetch limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we took Hasselmann’s (1974) form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates. More... »

PAGES

125-128

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-94-015-7717-5_16

DOI

http://dx.doi.org/10.1007/978-94-015-7717-5_16

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1026433143


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/02", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Physical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0299", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Other Physical Sciences", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE, De Bilt, The Netherlands", 
          "id": "http://www.grid.ac/institutes/grid.8653.8", 
          "name": [
            "Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE, De Bilt, The Netherlands"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Kamen", 
        "givenName": "G. J.", 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Max-Planck-Institut f\u00fcr Meteorologie, Bundesstrasse 55, Hamburg, Germany", 
          "id": "http://www.grid.ac/institutes/grid.450268.d", 
          "name": [
            "Max-Planck-Institut f\u00fcr Meteorologie, Bundesstrasse 55, Hamburg, Germany"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Hasselmann", 
        "givenName": "S.", 
        "id": "sg:person.013423560557.58", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013423560557.58"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Max-Planck-Institut f\u00fcr Meteorologie, Bundesstrasse 55, Hamburg, Germany", 
          "id": "http://www.grid.ac/institutes/grid.450268.d", 
          "name": [
            "Max-Planck-Institut f\u00fcr Meteorologie, Bundesstrasse 55, Hamburg, Germany"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Hasselmann", 
        "givenName": "K.", 
        "id": "sg:person.011507631147.31", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011507631147.31"
        ], 
        "type": "Person"
      }
    ], 
    "datePublished": "1985", 
    "datePublishedReg": "1985-01-01", 
    "description": "We consider the energy transfer equation for well developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parametrization proposed by Snyder et al (1981), which was based on their measurements in the Bight of Abaco, and which agrees with Miles\u2019 (1957, 1959) theory. The wave-wave interaction is computed with an algorithm given by Hasselmann et al (1984). The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters (cf. Hasselmann, 1974). Full details of this study are given elsewhere (Kamen, Hasselmann and Hasselmann, 1984). Here we summarize the main results. In the first part of our study we investigated whether the assumption that the equilibrium spectrum exists and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. This is illustrated in fig. 1, in which the assumed asymptotic spectrum and the corresponding source terms are given. The dissipation constant is chosen such that at any given frequency the total rate of change vanishes. As one can see there is no angular balance. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the study we chose a different approach. We assumed that the dissipation was given and we performed numerical experiments simulating fetch limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we took Hasselmann\u2019s (1974) form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates.", 
    "editor": [
      {
        "familyName": "Toba", 
        "givenName": "Yoshiaki", 
        "type": "Person"
      }, 
      {
        "familyName": "Mitsuyasu", 
        "givenName": "Hisashi", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-94-015-7717-5_16", 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-90-481-8415-6", 
        "978-94-015-7717-5"
      ], 
      "name": "The Ocean Surface", 
      "type": "Book"
    }, 
    "keywords": [
      "wave-wave interactions", 
      "asymptotic spectrum", 
      "source term", 
      "corresponding source terms", 
      "energy transfer equation", 
      "stationary solutions", 
      "asymptotic growth rate", 
      "wind input", 
      "unknown parameters", 
      "integral spectral parameters", 
      "dissipation parameter", 
      "energy balance equation", 
      "Pierson-Moskowitz spectrum", 
      "numerical experiments", 
      "transfer equation", 
      "equilibrium solution", 
      "dissipation function", 
      "balance equations", 
      "equilibrium spectrum", 
      "Bight of Abaco", 
      "wave spectrum", 
      "general assumptions", 
      "dissipation constants", 
      "equations", 
      "Hasselmann et al", 
      "ocean waves", 
      "dissipation balance", 
      "spectral parameters", 
      "angular distributions", 
      "et al", 
      "certain range", 
      "main results", 
      "full detail", 
      "factor coefficients", 
      "dissipation", 
      "standard type", 
      "solution results", 
      "solution", 
      "parameters", 
      "wind-sea spectrum", 
      "Hasselmann", 
      "assumption", 
      "parametrization", 
      "second part", 
      "vanishes", 
      "influence of wind", 
      "first part", 
      "theory", 
      "different approaches", 
      "waves", 
      "algorithm", 
      "spectra", 
      "terms", 
      "Fig. 1", 
      "existence", 
      "input", 
      "al", 
      "coefficient", 
      "frequency", 
      "distribution", 
      "wind", 
      "constants", 
      "conditions", 
      "total rate", 
      "results", 
      "function", 
      "approach", 
      "detail", 
      "interaction", 
      "Snyder et al", 
      "measurements", 
      "types", 
      "cases", 
      "experiments", 
      "growth rate", 
      "range", 
      "values", 
      "part", 
      "relation", 
      "balance", 
      "analysis", 
      "influence", 
      "rate", 
      "limited growth", 
      "study", 
      "Abaco", 
      "growth", 
      "changes", 
      "imbalance", 
      "Bight", 
      "miles"
    ], 
    "name": "The Fully Developed Wind-Sea Spectrum as a Solution of the Energy Balance Equation", 
    "pagination": "125-128", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1026433143"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-94-015-7717-5_16"
        ]
      }
    ], 
    "publisher": {
      "name": "Springer Nature", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-94-015-7717-5_16", 
      "https://app.dimensions.ai/details/publication/pub.1026433143"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2022-11-24T21:18", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20221124/entities/gbq_results/chapter/chapter_439.jsonl", 
    "type": "Chapter", 
    "url": "https://doi.org/10.1007/978-94-015-7717-5_16"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/978-94-015-7717-5_16'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-94-015-7717-5_16'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-94-015-7717-5_16'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-94-015-7717-5_16'


 

This table displays all metadata directly associated to this object as RDF triples.

171 TRIPLES      22 PREDICATES      116 URIs      109 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-94-015-7717-5_16 schema:about anzsrc-for:02
2 anzsrc-for:0299
3 schema:author N22a74fda3b2c4ddc95ce14ad774c4a1f
4 schema:datePublished 1985
5 schema:datePublishedReg 1985-01-01
6 schema:description We consider the energy transfer equation for well developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parametrization proposed by Snyder et al (1981), which was based on their measurements in the Bight of Abaco, and which agrees with Miles’ (1957, 1959) theory. The wave-wave interaction is computed with an algorithm given by Hasselmann et al (1984). The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters (cf. Hasselmann, 1974). Full details of this study are given elsewhere (Kamen, Hasselmann and Hasselmann, 1984). Here we summarize the main results. In the first part of our study we investigated whether the assumption that the equilibrium spectrum exists and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. This is illustrated in fig. 1, in which the assumed asymptotic spectrum and the corresponding source terms are given. The dissipation constant is chosen such that at any given frequency the total rate of change vanishes. As one can see there is no angular balance. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the study we chose a different approach. We assumed that the dissipation was given and we performed numerical experiments simulating fetch limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we took Hasselmann’s (1974) form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates.
7 schema:editor N0a5470e9dc8a48b7a9d9ac9728033d13
8 schema:genre chapter
9 schema:isAccessibleForFree false
10 schema:isPartOf N2037d271203a4fbdb71d18d5e22b7bd2
11 schema:keywords Abaco
12 Bight
13 Bight of Abaco
14 Fig. 1
15 Hasselmann
16 Hasselmann et al
17 Pierson-Moskowitz spectrum
18 Snyder et al
19 al
20 algorithm
21 analysis
22 angular distributions
23 approach
24 assumption
25 asymptotic growth rate
26 asymptotic spectrum
27 balance
28 balance equations
29 cases
30 certain range
31 changes
32 coefficient
33 conditions
34 constants
35 corresponding source terms
36 detail
37 different approaches
38 dissipation
39 dissipation balance
40 dissipation constants
41 dissipation function
42 dissipation parameter
43 distribution
44 energy balance equation
45 energy transfer equation
46 equations
47 equilibrium solution
48 equilibrium spectrum
49 et al
50 existence
51 experiments
52 factor coefficients
53 first part
54 frequency
55 full detail
56 function
57 general assumptions
58 growth
59 growth rate
60 imbalance
61 influence
62 influence of wind
63 input
64 integral spectral parameters
65 interaction
66 limited growth
67 main results
68 measurements
69 miles
70 numerical experiments
71 ocean waves
72 parameters
73 parametrization
74 part
75 range
76 rate
77 relation
78 results
79 second part
80 solution
81 solution results
82 source term
83 spectra
84 spectral parameters
85 standard type
86 stationary solutions
87 study
88 terms
89 theory
90 total rate
91 transfer equation
92 types
93 unknown parameters
94 values
95 vanishes
96 wave spectrum
97 wave-wave interactions
98 waves
99 wind
100 wind input
101 wind-sea spectrum
102 schema:name The Fully Developed Wind-Sea Spectrum as a Solution of the Energy Balance Equation
103 schema:pagination 125-128
104 schema:productId N91de423625cc4593a5a4e1cbdb456443
105 Nf0237998749b40708add5829fed6d0c0
106 schema:publisher N31d5638d87df46e4a89d1cee5c7bce69
107 schema:sameAs https://app.dimensions.ai/details/publication/pub.1026433143
108 https://doi.org/10.1007/978-94-015-7717-5_16
109 schema:sdDatePublished 2022-11-24T21:18
110 schema:sdLicense https://scigraph.springernature.com/explorer/license/
111 schema:sdPublisher N13a9aebc0ae94f06813820ec1a6ccffe
112 schema:url https://doi.org/10.1007/978-94-015-7717-5_16
113 sgo:license sg:explorer/license/
114 sgo:sdDataset chapters
115 rdf:type schema:Chapter
116 N0a5470e9dc8a48b7a9d9ac9728033d13 rdf:first Nc6d15549156142a18fa98d33bb46fc01
117 rdf:rest Nc4c8fb50b4b247a1938b5dab7a96f158
118 N13a9aebc0ae94f06813820ec1a6ccffe schema:name Springer Nature - SN SciGraph project
119 rdf:type schema:Organization
120 N2037d271203a4fbdb71d18d5e22b7bd2 schema:isbn 978-90-481-8415-6
121 978-94-015-7717-5
122 schema:name The Ocean Surface
123 rdf:type schema:Book
124 N22a74fda3b2c4ddc95ce14ad774c4a1f rdf:first Na36c8e8aa5c8422f834e79ded80685dc
125 rdf:rest Ne473a5e436da4db991e10cc83ca7986a
126 N31d5638d87df46e4a89d1cee5c7bce69 schema:name Springer Nature
127 rdf:type schema:Organisation
128 N60a43988c8f84ce49273522f50152c3f rdf:first sg:person.011507631147.31
129 rdf:rest rdf:nil
130 N85d26a42cadf4cefac9aa309973c6e1d schema:familyName Mitsuyasu
131 schema:givenName Hisashi
132 rdf:type schema:Person
133 N91de423625cc4593a5a4e1cbdb456443 schema:name doi
134 schema:value 10.1007/978-94-015-7717-5_16
135 rdf:type schema:PropertyValue
136 Na36c8e8aa5c8422f834e79ded80685dc schema:affiliation grid-institutes:grid.8653.8
137 schema:familyName Kamen
138 schema:givenName G. J.
139 rdf:type schema:Person
140 Nc4c8fb50b4b247a1938b5dab7a96f158 rdf:first N85d26a42cadf4cefac9aa309973c6e1d
141 rdf:rest rdf:nil
142 Nc6d15549156142a18fa98d33bb46fc01 schema:familyName Toba
143 schema:givenName Yoshiaki
144 rdf:type schema:Person
145 Ne473a5e436da4db991e10cc83ca7986a rdf:first sg:person.013423560557.58
146 rdf:rest N60a43988c8f84ce49273522f50152c3f
147 Nf0237998749b40708add5829fed6d0c0 schema:name dimensions_id
148 schema:value pub.1026433143
149 rdf:type schema:PropertyValue
150 anzsrc-for:02 schema:inDefinedTermSet anzsrc-for:
151 schema:name Physical Sciences
152 rdf:type schema:DefinedTerm
153 anzsrc-for:0299 schema:inDefinedTermSet anzsrc-for:
154 schema:name Other Physical Sciences
155 rdf:type schema:DefinedTerm
156 sg:person.011507631147.31 schema:affiliation grid-institutes:grid.450268.d
157 schema:familyName Hasselmann
158 schema:givenName K.
159 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.011507631147.31
160 rdf:type schema:Person
161 sg:person.013423560557.58 schema:affiliation grid-institutes:grid.450268.d
162 schema:familyName Hasselmann
163 schema:givenName S.
164 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.013423560557.58
165 rdf:type schema:Person
166 grid-institutes:grid.450268.d schema:alternateName Max-Planck-Institut für Meteorologie, Bundesstrasse 55, Hamburg, Germany
167 schema:name Max-Planck-Institut für Meteorologie, Bundesstrasse 55, Hamburg, Germany
168 rdf:type schema:Organization
169 grid-institutes:grid.8653.8 schema:alternateName Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE, De Bilt, The Netherlands
170 schema:name Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE, De Bilt, The Netherlands
171 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...