On the Solution of Long's Equation with Shear - DigitalCommons@WPI
Worcester Polytechnic Institute
DigitalCommons@WPI Mathematical Sciences Faculty Publications
Department of Mathematical Sciences
8-1-2006
On the Solution of Long's Equation with Shear Mayer Humi Worcester Polytechnic Institute, [email protected]
Follow this and additional works at: http://digitalcommons.wpi.edu/mathematicalsciences-pubs Part of the Mathematics Commons Suggested Citation Humi, Mayer (2006). On the Solution of Long's Equation with Shear. SIAM Journal on Applied Mathematics, 66(6), 1839-1852. Retrieved from: http://digitalcommons.wpi.edu/mathematicalsciences-pubs/23
This Article is brought to you for free and open access by the Department of Mathematical Sciences at DigitalCommons@WPI. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of DigitalCommons@WPI.
SIAM J. APPL. MATH. Vol. 66, No. 6, pp. 1839-1852
(c) 2006 Society forIndustrial and Applied Mathematics
ON THE SOLUTION OF LONG'S EQUATION WITH SHEAR* MAYER HUMI+ Abstract. Long'sequationdescribestwodimensional stratified flowoverterrain.Its numerical wereinvestigated solutionsundervariousapproximations by manyauthorsunderthe assumption thatthebase flowfieldis withoutshear.Specialattentionwas paid to theproperties ofthegravity wavesthat are predictedto be generatedas a result. In this paper we address,analytically, the natureand properties of thesesolutionswhenshearis presentand derivesomeconstraints on the ofgravitywavesunderthesecircumstances. possiblegeneration waves,Long'sequation,shear Key words, gravity AMS subject classifications. 76B60,76E05,76E30,86A10 DOI. 10.1137/050627794
in1. Introduction.Long'sequation[1, 2, 3, 4] modelsthe flowofstratified in two dimensions over terfluid the Boussinesqapproximation) compressible (in flowfieldfaruprain. Whenthe base stateofthe flow(thatis, the unperturbed the form of solutions(in shear,thenumerical steadylee waves)of stream)is without werestudiedby manyauthors and approximations thisequationin varioussettings in thesestudieswas [5,6, 7, 8, 9, 10, 11, 12, 13]. The mostcommonapproximation or a function overthe computato a constant Vaisala frequency to set Bruntstep and which thevaluesoftheparameters tionaldomain.Moreover, (3 appearin this /x nonlinear terms and oneofthe equationweresetto zero.In this(singular)limitthe the reduces to intheequationdropoutand equation leadingsecondorderderivatives domain. studies Careful overa twodimensional oscillator thatofa linearharmonic [8] in the is present arejustified unlesswavebreaking showedthattheseapproximations solution[9]. framework fortheanalysisofexperithetheoretical Long'sequationalsoprovides baseflow.(An assumption ofshearless mentaldata [15,16,17]undertheassumption which,in general,is not supportedby the data.) An extensivelist of references appearsin [18,19,20]. wasinitiated An analyticapproachto thestudyofthisequationanditssolutions shearand under bytheauthor[14].We showedthatfora base flowwithout recently Wealso termsintheequationcanbe simplified. thenonlinear rathermildrestrictions in thisequation oscillations thenonlinear the "slowvariable"thatcontrols identified fortheattenuation of deriveda formula approximation, and,usingphaseaveraging relatedto the withheight.Thisresultis generically thestreamfunction perturbation termsin Long'sequation. ofthenonlinear presence The objectiveof thispaperis to studythe natureof the solutionsto Long's Vaisala frequency is a equationwhenshearis presentin the base flowand Bruntwhichdependsolelyon thebase flow ofheight.Usingconditions function continuous natureoftheperturbathequalitative we characterize Vaisalafrequency and Bruntvarieswithheight.Theseresults tionsfromthebase flowand howtheiramplitude *Received theeditorsMarch27, 2005;acceptedforpublication(in revisedform)May22, 2006; by August22, 2006. publishedelectronically http://www.siam.org/journals/siap/66-6/62779.html tDepartmentof MathematicalSciences,WorcesterPolytechnicInstitute,100 InstituteRoad, MA 01609([email protected]). Worcester, 1839
1840
MAYER HUMI
axe independent ofthe actualdetaileddescription ofthe terrainthatcausedthese whichtheseperturbations Furthermore we derive conditions under perturbations. are notoscillatory; no waves are i.e., gravity by theflow.To thebestof generated ourknowledge in theliterature before(in thecontext thisissuewasneverconsidered ofLong'sequation). The planofthepaperis as follows: In section2 we present a shortreviewofthe derivation ofLong'sequationand thesolutionofits linearized version.In section3 we deriveconstraints on the solutionsof thisequationin a generalsettingand in in the ofshear.In section4 solutions tothisequationwithdifferent particular presence shearprofiles ofLong's are studiedexplicitly. In section5 we carryout simulations for in and shear less and shear base section with flows. We end 6 equation summary conclusions. 2. Long's equation: A short review. In twodimensions (x,z) theflowofa inviscid and stratified fluid the steady incompressible Boussinesqapproximation) (in is modeledbythefollowing equations: (2.1)
ux+wz = 0,
(2.2)
upx+ wpz= 0,
(2.3)
p(uux+ wuz) = -pxi
(2.4)
p(uwx+ wwz)= -pz - pg,
wheresubscripts indicatedifferentiation withrespectto theindicatedvariable,u = and g is theacceleration p is thepressure, (m,w) is thefluidvelocity, p is itsdensity, ofgravity. We can nondimensionalize theseequationsbyintroducing _ (2-5)
x
_ No
_
u
LNo
-p=Pp=^p, PO 9UoPo
whereL represents a characteristic thefree lengthand £/o> Po represent, respectively, streamvelocity and density. BruntVaisalafrequency No is thecharacteristic
(2-6)
N>= -i- £>. dz Po
In thesenewvariables,(2.1)-(2.4) takethefollowing form(forbrevity we drop thebars): (2.7)
ux+wx= 0,
(2.8)
upx+ wpz= 0,
(2.9)
Pp(uux+ wuz) = -pz,
(2.10)
+ wwz)= -n~2(pz + p), f3p{uwx
LONG'S EQUATION WITH SHEAR
1841
where
(2.11)
0=^,
effects(assuming 0 is the Boussinesq parameter[13] which controlsstratification J70zfz0), and //is the long wave parameterwhichcontrolsdispersiveeffects(or the deviationfromthe hydrostaticapproximation).In the limit[i = 0 the hydrostatic approximationis fullysatisfied[20]. In view of (2.7) we can introducea streamfunctioni\)so that u = i/jz1w = -t/>x.
(2.13)
Prom(2.8) and (2.13) we inferthat p = p(ip) and (aftersome algebra) derivethe equation forip [13]: following
(2.14)
- N2(l>) + = + *„ + /xVxx MV2)]
DigitalCommons@WPI Mathematical Sciences Faculty Publications
Department of Mathematical Sciences
8-1-2006
On the Solution of Long's Equation with Shear Mayer Humi Worcester Polytechnic Institute, [email protected]
Follow this and additional works at: http://digitalcommons.wpi.edu/mathematicalsciences-pubs Part of the Mathematics Commons Suggested Citation Humi, Mayer (2006). On the Solution of Long's Equation with Shear. SIAM Journal on Applied Mathematics, 66(6), 1839-1852. Retrieved from: http://digitalcommons.wpi.edu/mathematicalsciences-pubs/23
This Article is brought to you for free and open access by the Department of Mathematical Sciences at DigitalCommons@WPI. It has been accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of DigitalCommons@WPI.
SIAM J. APPL. MATH. Vol. 66, No. 6, pp. 1839-1852
(c) 2006 Society forIndustrial and Applied Mathematics
ON THE SOLUTION OF LONG'S EQUATION WITH SHEAR* MAYER HUMI+ Abstract. Long'sequationdescribestwodimensional stratified flowoverterrain.Its numerical wereinvestigated solutionsundervariousapproximations by manyauthorsunderthe assumption thatthebase flowfieldis withoutshear.Specialattentionwas paid to theproperties ofthegravity wavesthat are predictedto be generatedas a result. In this paper we address,analytically, the natureand properties of thesesolutionswhenshearis presentand derivesomeconstraints on the ofgravitywavesunderthesecircumstances. possiblegeneration waves,Long'sequation,shear Key words, gravity AMS subject classifications. 76B60,76E05,76E30,86A10 DOI. 10.1137/050627794
in1. Introduction.Long'sequation[1, 2, 3, 4] modelsthe flowofstratified in two dimensions over terfluid the Boussinesqapproximation) compressible (in flowfieldfaruprain. Whenthe base stateofthe flow(thatis, the unperturbed the form of solutions(in shear,thenumerical steadylee waves)of stream)is without werestudiedby manyauthors and approximations thisequationin varioussettings in thesestudieswas [5,6, 7, 8, 9, 10, 11, 12, 13]. The mostcommonapproximation or a function overthe computato a constant Vaisala frequency to set Bruntstep and which thevaluesoftheparameters tionaldomain.Moreover, (3 appearin this /x nonlinear terms and oneofthe equationweresetto zero.In this(singular)limitthe the reduces to intheequationdropoutand equation leadingsecondorderderivatives domain. studies Careful overa twodimensional oscillator thatofa linearharmonic [8] in the is present arejustified unlesswavebreaking showedthattheseapproximations solution[9]. framework fortheanalysisofexperithetheoretical Long'sequationalsoprovides baseflow.(An assumption ofshearless mentaldata [15,16,17]undertheassumption which,in general,is not supportedby the data.) An extensivelist of references appearsin [18,19,20]. wasinitiated An analyticapproachto thestudyofthisequationanditssolutions shearand under bytheauthor[14].We showedthatfora base flowwithout recently Wealso termsintheequationcanbe simplified. thenonlinear rathermildrestrictions in thisequation oscillations thenonlinear the "slowvariable"thatcontrols identified fortheattenuation of deriveda formula approximation, and,usingphaseaveraging relatedto the withheight.Thisresultis generically thestreamfunction perturbation termsin Long'sequation. ofthenonlinear presence The objectiveof thispaperis to studythe natureof the solutionsto Long's Vaisala frequency is a equationwhenshearis presentin the base flowand Bruntwhichdependsolelyon thebase flow ofheight.Usingconditions function continuous natureoftheperturbathequalitative we characterize Vaisalafrequency and Bruntvarieswithheight.Theseresults tionsfromthebase flowand howtheiramplitude *Received theeditorsMarch27, 2005;acceptedforpublication(in revisedform)May22, 2006; by August22, 2006. publishedelectronically http://www.siam.org/journals/siap/66-6/62779.html tDepartmentof MathematicalSciences,WorcesterPolytechnicInstitute,100 InstituteRoad, MA 01609([email protected]). Worcester, 1839
1840
MAYER HUMI
axe independent ofthe actualdetaileddescription ofthe terrainthatcausedthese whichtheseperturbations Furthermore we derive conditions under perturbations. are notoscillatory; no waves are i.e., gravity by theflow.To thebestof generated ourknowledge in theliterature before(in thecontext thisissuewasneverconsidered ofLong'sequation). The planofthepaperis as follows: In section2 we present a shortreviewofthe derivation ofLong'sequationand thesolutionofits linearized version.In section3 we deriveconstraints on the solutionsof thisequationin a generalsettingand in in the ofshear.In section4 solutions tothisequationwithdifferent particular presence shearprofiles ofLong's are studiedexplicitly. In section5 we carryout simulations for in and shear less and shear base section with flows. We end 6 equation summary conclusions. 2. Long's equation: A short review. In twodimensions (x,z) theflowofa inviscid and stratified fluid the steady incompressible Boussinesqapproximation) (in is modeledbythefollowing equations: (2.1)
ux+wz = 0,
(2.2)
upx+ wpz= 0,
(2.3)
p(uux+ wuz) = -pxi
(2.4)
p(uwx+ wwz)= -pz - pg,
wheresubscripts indicatedifferentiation withrespectto theindicatedvariable,u = and g is theacceleration p is thepressure, (m,w) is thefluidvelocity, p is itsdensity, ofgravity. We can nondimensionalize theseequationsbyintroducing _ (2-5)
x
_ No
_
u
LNo
-p=Pp=^p, PO 9UoPo
whereL represents a characteristic thefree lengthand £/o> Po represent, respectively, streamvelocity and density. BruntVaisalafrequency No is thecharacteristic
(2-6)
N>= -i- £>. dz Po
In thesenewvariables,(2.1)-(2.4) takethefollowing form(forbrevity we drop thebars): (2.7)
ux+wx= 0,
(2.8)
upx+ wpz= 0,
(2.9)
Pp(uux+ wuz) = -pz,
(2.10)
+ wwz)= -n~2(pz + p), f3p{uwx
LONG'S EQUATION WITH SHEAR
1841
where
(2.11)
0=^,
effects(assuming 0 is the Boussinesq parameter[13] which controlsstratification J70zfz0), and //is the long wave parameterwhichcontrolsdispersiveeffects(or the deviationfromthe hydrostaticapproximation).In the limit[i = 0 the hydrostatic approximationis fullysatisfied[20]. In view of (2.7) we can introducea streamfunctioni\)so that u = i/jz1w = -t/>x.
(2.13)
Prom(2.8) and (2.13) we inferthat p = p(ip) and (aftersome algebra) derivethe equation forip [13]: following
(2.14)
- N2(l>) + = + *„ + /xVxx MV2)]
