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disp_eig.m
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disp_eig.m
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function out= disp_eig(yv,oper)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% DISPERSION -- Plasma dispersion characteristics
% Part of the RAYcON package
%
% yin Position in phase space [r,z,k_r,k_z]
% oper
% 'Msw' Dispersion relation approximated 1x1
% 'Dsp' Disperison relation according to MODEL
% 'Ant' Antenna IC
% 'Trj' RHS for trajectory
% 'Sgn' Sign of dt/dsigma, set by dUdom
% 'Pol' Polarization vector, eigenvector of D matrix
%
% ToDo Optimize computing
% - change Dij(c) vectors into D(c,i,j) matrix
% - remove terms that are zero using emacs search/substitute
%
% A. JAUN, Alfven Laboratory, KTH, 100 44 Stockholm, Sweden
% A.N. KAUFMAN, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA
% E.R. TRACY, College of William & Mary, Williamsburg, VA 23187-8795, USA
%
% (C) Version 7.0, 14-Aug-2006. All Rights Reserved.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
global cnst
global plasma rays
i = complex(0,1);
isCmplx=0;
%
% ----- Vectorized input quantities ------------------------------------------
%
if numel(yv) == 4
yin = yv.';
sym_mat = eye(4);
eval2nd = 0;
else
yin = yv;
% now assume one ray, and tracing position and symplectic matrix
sym_mat = reshape(yin(5:20),4,4);
yin = yin(1:4).';
eval2nd = 1;
end
if strcmp(oper,'Trj')
eval1st = 1;
else
eval1st = 0;
eval2nd = 0;
end
%
% ----- Adjusted position & wave vector in cylindrical coordinates -----------
zro=zeros(size(yin(:,1))); one=ones(size(zro));
if (rays.odeDim==4||rays.odeDim>=9)
rr=yin(:,1); zz=yin(:,2); kr=yin(:,3); kz=yin(:,4);
kf=one.*plasma.kant(2); % fi=omega*tim/(kf*rr);
end
rho=sqrt((rr-plasma.r0).^2+zz.^2);
theta=atan2(plasma.r0-rr,-zz)+pi/2;
%
%
% ===== Local plasma parameters ==============================================
%
% ----- Magnetic field and topology ------------------------------------------
[b,dbds,dbdt,bp,sflx,dsdr,dsdz,dtdr,dtdz, ...
ener,enef,enez,eber,ebef,ebez,eper,epef,epez, ...
dbds2,dbdst,dbdt2, ...
dsdr2,dsdrz,dsdz2,dsdr3,dsdr2z,dsdrz2,dsdz3, ...
dtdr2,dtdrz,dtdz2,...
denerdr, denefdr, denezdr,...
denerdz, denefdz, denezdz,...
deberdr, debefdr, debezdr,...
deberdz, debefdz, debezdz,...
deperdr, depefdr, depezdr,...
deperdz, depefdz, depezdz,... % end of first derivs
denerdr2, denerdrz, denerdz2,...
denefdr2, denefdrz, denefdz2,...
denezdr2, denezdrz, denezdz2,...
deberdr2, deberdrz, deberdz2,...
debefdr2, debefdrz, debefdz2,...
debezdr2, debezdrz, debezdz2,...
deperdr2, deperdrz, deperdz2,...
depefdr2, depefdrz, depefdz2,...
depezdr2, depezdrz, depezdz2]=magnetic(rho,theta);
%
% ----- Wave vector and refraction index -------------------------------------
coom=cnst.c/plasma.omega; coomsq=coom^2; om2=plasma.omega^2;
kn=kr.*ener +kf.*enef +kz.*enez; Nn=coom*kn; Nn2=Nn.^2;
kb=kr.*eber +kf.*ebef +kz.*ebez; Nb=coom*kb; Nb2=Nb.^2;
kp=kr.*eper +kf.*epef +kz.*epez; Np=coom*kp; Np2=Np.^2; %N2=Nn2+Nb2+Np2;
Nr = coom*kr;
Nf = coom*kf;
Nz = coom*kz;
%
% ----- Parabolic profiles and logarithmic derivatives -----------------------
nspec=size(plasma.amass,2);
p=(1-sflx.^2*plasma.na);
dLNpds = zeros(length(sflx),nspec);
dLNpds2 = dLNpds;
n = dLNpds;
T = dLNpds;
for k=1:nspec
dLNpds(:,k) =-2*sflx.*plasma.na(k).*plasma.nb(k)./p(:,k);
dLNpds2(:,k)=dLNpds(:,k)./sflx-dLNpds(:,k).^2./plasma.nb(k);
n(:,k)=plasma.n0(k).*p(:,k).^plasma.nb(k); % Density
T(:,k)=plasma.t0(k).*(1-sflx.^2*plasma.ta(k)).^plasma.nb(k); % Temperature
end
dLNnds =dLNpds;
dLNnds2=dLNpds2;
%
% ----- Plasma parameters and logarithmic derivatives ------------------------
omp2= ones(size(theta))*((plasma.acharge'.*cnst.e).^2./...
(plasma.amass'.*cnst.mp*cnst.eps0))'.*n;
omp = sqrt(omp2); % Plasma frequencies
omc = b*((plasma.acharge'.*cnst.e)./(plasma.amass'.*cnst.mp))'; % Cyclotron frequencies
omc2= omc.^2;
caoc2 = 1./sum(omp2./omc2, 2); % (c_Alfven/c_light)^2
dLNomp2ds=dLNnds; % Logarithmic derivatives
one=ones(size(plasma.amass'))';
dLNomcds =(dbds./b)*one;
dLNomcdt =(dbdt./b)*one;
% second derivatives
dLNomcds2=(dbds2./b-(dbds.^2 )./(b.^2))*one;
dLNomcdst=(dbdst./b-(dbds.*dbdt)./(b.^2))*one;
dLNomcdt2=(dbdt2./b-(dbdt.^2 )./(b.^2))*one;
switch plasma.MODEL(1:6)
%
% +++++ Cold plasma model ++++++++++++++++++++++++++++++++++++++++++++++++++++
%
case {'cld2x2'}
% --- Elementary functions and derivatives ---------------------------------
if (isCmplx && ~eval1st)
iomceps=0.003*i*om2;
else
iomceps=0;
end
omc2Mom2= omc2-om2-2*iomceps; % S,D,P
Si=omp2./omc2Mom2; S=1+sum( Si ,2);
Di=(omc/plasma.omega).*Si; D= sum( Di ,2);
% Pi=omp2/om2; P=1-sum( Pi ,2);
% 1st derivatives S,D,P
zro=zeros(size(S));
dLNSids =dLNnds-2.*omc2./omc2Mom2.*dLNomcds; dSds = sum( Si.*dLNSids ,2);
dLNSidt = -2.*omc2./omc2Mom2.*dLNomcdt; dSdt = sum( Si.*dLNSidt ,2);
dLNSidom =2*plasma.omega./omc2Mom2; dSdom= sum( Si.*dLNSidom,2);
dLNDids =dLNSids + dLNomcds; dDds = sum( Di.*dLNDids ,2);
dLNDidt =dLNSidt + dLNomcdt; dDdt = sum( Di.*dLNDidt ,2);
dLNDidom =(3*om2-omc2)./(plasma.omega*omc2Mom2); dDdom= sum( Di.*dLNDidom,2);
% % These are only needed in the 3x3 model
% zroi=zeros(size(Si));
% dLNPids =dLNnds;
% dPds =-sum( Pi.*dLNPids ,2);
% dLNPidt =zroi;
% dPdt = zro;
% dLNPidom=2/plasma.omega;
% dPdom= sum( Pi.*dLNPidom,2);
% second derivatives
if eval2nd
omOM=om2 ./omc2Mom2;
ocOM=omc2./omc2Mom2;
dLNSids2= 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcds -dLNomcds2)+dLNnds2;
dLNSidst= 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcdt -dLNomcdst);
dLNSidt2= 2*ocOM.*(2*omOM.*dLNomcdt.*dLNomcdt -dLNomcdt2);
dLNDids2= dLNomcds2 +dLNSids2;
dLNDidst= dLNomcdst +dLNSidst;
dLNDidt2= dLNomcdt2 +dLNSidt2;
dSids2 = Si.*(dLNSids2 +dLNSids.*dLNSids); dSds2=sum( dSids2 ,2);
dSidst = Si.*(dLNSidst +dLNSids.*dLNSidt); dSdst=sum( dSidst ,2);
dSidt2 = Si.*(dLNSidt2 +dLNSidt.*dLNSidt); dSdt2=sum( dSidt2 ,2);
dDids2 = Di.*(dLNDids2 +dLNDids.*dLNDids); dDds2=sum( dDids2 ,2);
dDidst = Di.*(dLNDidst +dLNDids.*dLNDidt); dDdst=sum( dDidst ,2);
dDidt2 = Di.*(dLNDidt2 +dLNDidt.*dLNDidt); dDdt2=sum( dDidt2 ,2);
% % These are only needed in the 3x3 model
% dLNPids2= dLNnds2; dLNPidst= zroi; dLNPidt2= zroi;
% dPids2= Pi.*(dLNPids2 +dLNPids.*dLNPids); dPds2=sum( dPids2 ,2);
% dPidst= Pi.*(dLNPidst +dLNPids.*dLNPidt); dPdst=sum( dPidst ,2);
% dPidt2= Pi.*(dLNPidt2 +dLNPidt.*dLNPidt); dPdt2=sum( dPidt2 ,2);
end
% --- Tensor elements and derivatives --------------------------------------
D11 = Nb2+Np2-S;
D12 =-Nn.*Nb-D*i;
D22 = Nn2+Np2-S;
cD12 = conj(D12);
% Dispersion tensor
DD=[D11 D12 cD12 D22];
% We will assume that the input yv is size [1,4], then reshape works
DD=reshape(DD,2,2);
[evects,evals]=eig(DD); % Two values
evs = diag(evals);
ind = find(abs(evs)==min(abs(evs)));
U = evs(ind); % The eval nearest zero is the one we want
pol = evects(:,ind); % get the column with the right evector
% Conversion monitor
mon2 = abs(D11 + D22); % trace DD
if eval1st
dD11ds =-dSds; dD11dt =-dSdt;
dD12ds =-dDds*i; dD12dt =-dDdt*i;
dD22ds =-dSds; dD22dt =-dSdt;
dD11dkn= zro; dD11dkb= Nb*coom*2; dD11dkp= Np*coom*2;
dD12dkn=-Nb*coom; dD12dkb=-Nn*coom; dD12dkp= zro;
dD22dkn= Nn*coom*2; dD22dkb= zro; dD22dkp= Np*coom*2;
dD11dkr=dD11dkn.*ener +dD11dkb.*eber +dD11dkp.*eper;
dD11dkz=dD11dkn.*enez +dD11dkb.*ebez +dD11dkp.*epez;
dD12dkr=dD12dkn.*ener +dD12dkb.*eber +dD12dkp.*eper;
dD12dkz=dD12dkn.*enez +dD12dkb.*ebez +dD12dkp.*epez;
dD22dkr=dD22dkn.*ener +dD22dkb.*eber +dD22dkp.*eper;
dD22dkz=dD22dkn.*enez +dD22dkb.*ebez +dD22dkp.*epez;
dD11dom=-2/plasma.omega*(Nb2+Np2)-dSdom;
dD12dom=-2/plasma.omega*(Nn.*Nb) -dDdom*i;
dD22dom=-2/plasma.omega*(Nn2+Np2)-dSdom;
% Sub-blocks of (dispersion matrix - eigenval*id)
subsum = D11+D22-2*U;
% Convert to r, z coordinates.
% Note that there are extra correction terms due to field
% curvature. They are derivatives of the trasformation elements:
% (d/dr)(ebef) etc.
dNndr = Nr.*denerdr + Nf.*denefdr + Nz.*denezdr;
dNndz = Nr.*denerdz + Nf.*denefdz + Nz.*denezdz;
dNbdr = Nr.*deberdr + Nf.*debefdr + Nz.*debezdr;
dNbdz = Nr.*deberdz + Nf.*debefdz + Nz.*debezdz;
dNpdr = Nr.*deperdr + Nf.*depefdr + Nz.*depezdr;
dNpdz = Nr.*deperdz + Nf.*depefdz + Nz.*depezdz;
dD11dr = dD11ds.*dsdr + dD11dt.*dtdr + 2*(Nb.*dNbdr + Np.*dNpdr);
dD11dz = dD11ds.*dsdz + dD11dt.*dtdz + 2*(Nb.*dNbdz + Np.*dNpdz);
dD12dr = dD12ds.*dsdr + dD12dt.*dtdr - Nn.*dNbdr - Nb.*dNndr;
dD12dz = dD12ds.*dsdz + dD12dt.*dtdz - Nn.*dNbdz - Nb.*dNndz;
dD22dr = dD22ds.*dsdr + dD22dt.*dtdr + 2*(Nn.*dNndr + Np.*dNpdr);
dD22dz = dD22ds.*dsdz + dD22dt.*dtdz + 2*(Nn.*dNndz + Np.*dNpdz);
% Gather things together to do all derivatives at the same time
dD11_vec = [dD11dom dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dom dD12dr dD12dz dD12dkr dD12dkz];
dD22_vec = [dD22dom dD22dr dD22dz dD22dkr dD22dkz];
% Derivatives of the eigenvalue
%dU_vec = [dUdom dUds dUdt dUdkn dUdkb dUdkp];
dU_vec = (dD11_vec.*(D22-U) +dD22_vec.*(D11-U))./subsum...
- 2*real(cD12.*dD12_vec)./subsum;
end
% terms needed for second derivative of hamiltonian
if eval2nd
dSdr2 = dsdr2.*dSds + dtdr2.*dSdt ...
+ dsdr.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdr.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdrz = dsdrz.*dSds + dtdrz.*dSdt ...
+ dsdz.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdz.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdz2 = dsdz2.*dSds + dtdz2.*dSdt ...
+ dsdz.*(dsdz.*dSds2 + dtdz.*dSdst) + dtdz.*(dsdz.*dSdst + dtdz.*dSdt2);
dDdr2 = dsdr2.*dDds + dtdr2.*dDdt ...
+ dsdr.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdr.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdrz = dsdrz.*dDds + dtdrz.*dDdt ...
+ dsdz.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdz.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdz2 = dsdz2.*dDds + dtdz2.*dDdt ...
+ dsdz.*(dsdz.*dDds2 + dtdz.*dDdst) + dtdz.*(dsdz.*dDdst + dtdz.*dDdt2);
dNndr2 = Nr.*denerdr2 + Nf.*denefdr2 + Nz.*denezdr2;
dNndrz = Nr.*denerdrz + Nf.*denefdrz + Nz.*denezdrz;
dNndz2 = Nr.*denerdz2 + Nf.*denefdz2 + Nz.*denezdz2;
dNbdr2 = Nr.*deberdr2 + Nf.*debefdr2 + Nz.*debezdr2;
dNbdrz = Nr.*deberdrz + Nf.*debefdrz + Nz.*debezdrz;
dNbdz2 = Nr.*deberdz2 + Nf.*debefdz2 + Nz.*debezdz2;
dNpdr2 = Nr.*deperdr2 + Nf.*depefdr2 + Nz.*depezdr2;
dNpdrz = Nr.*deperdrz + Nf.*depefdrz + Nz.*depezdrz;
dNpdz2 = Nr.*deperdz2 + Nf.*depefdz2 + Nz.*depezdz2;
dD11dr2=2*(dNbdr.*dNbdr + Nb.*dNbdr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD11drz=2*(dNbdr.*dNbdz + Nb.*dNbdrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD11dz2=2*(dNbdz.*dNbdz + Nb.*dNbdz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD12dr2=-(Nn.*dNbdr2 + 2*dNndr.*dNbdr + Nb.*dNndr2)-i*dDdr2;
dD12drz=-(Nn.*dNbdrz + dNndr.*dNbdz + dNndz.*dNbdr + Nb.*dNndrz)-i*dDdrz;
dD12dz2=-(Nn.*dNbdz2 + 2*dNndz.*dNbdz + Nb.*dNndz2)-i*dDdz2;
dD22dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD22drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD22dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD11drkr = 2*coom*(eber.*dNbdr + Nb.*deberdr + eper.*dNpdr + Np.*deperdr);
dD11drkz = 2*coom*(ebez.*dNbdr + Nb.*debezdr + epez.*dNpdr + Np.*depezdr);
dD11dzkr = 2*coom*(eber.*dNbdz + Nb.*deberdz + eper.*dNpdz + Np.*deperdz);
dD11dzkz = 2*coom*(ebez.*dNbdz + Nb.*debezdz + epez.*dNpdz + Np.*depezdz);
dD12drkr = -coom*(denerdr.*Nb + dNndr.*eber + ener.*dNbdr + Nn.*deberdr);
dD12drkz = -coom*(denezdr.*Nb + dNndr.*ebez + enez.*dNbdr + Nn.*debezdr);
dD12dzkr = -coom*(denerdz.*Nb + dNndz.*eber + ener.*dNbdz + Nn.*deberdz);
dD12dzkz = -coom*(denezdz.*Nb + dNndz.*ebez + enez.*dNbdz + Nn.*debezdz);
dD22drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eper.*dNpdr + Np.*deperdr);
dD22drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + epez.*dNpdr + Np.*depezdr);
dD22dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eper.*dNpdz + Np.*deperdz);
dD22dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + epez.*dNpdz + Np.*depezdz);
dD11dkr2 = 2*coomsq*(eber.^2 + eper.^2);
dD11dkz2 = 2*coomsq*(ebez.^2 + epez.^2);
dD11dkrkz= 2*coomsq*(eber.*ebez + eper.*epez);
dD12dkr2 = -coomsq*(ener.*eber + ener.*eber);
dD12dkz2 = -coomsq*(ener.*ebez + enez.*eber);
dD12dkrkz= -coomsq*(enez.*ebez + enez.*ebez);
dD22dkr2 = 2*coomsq*(ener.^2 + eper.^2);
dD22dkz2 = 2*coomsq*(enez.^2 + epez.^2);
dD22dkrkz= 2*coomsq*(ener.*enez + eper.*epez);
% dD11_mat = [dr dz dkr dkz].'*[dr dz dkr dkz];
dD11_mat = [dD11dr2 dD11drz dD11drkr dD11drkz;
dD11drz dD11dz2 dD11dzkr dD11dzkz;
dD11drkr dD11dzkr dD11dkr2 dD11dkrkz;
dD11drkz dD11dzkz dD11dkrkz dD11dkz2 ];
dD12_mat = [dD12dr2 dD12drz dD12drkr dD12drkz;
dD12drz dD12dz2 dD12dzkr dD12dzkz;
dD12drkr dD12dzkr dD12dkr2 dD12dkrkz;
dD12drkz dD12dzkz dD12dkrkz dD12dkz2 ];
dD22_mat = [dD22dr2 dD22drz dD22drkr dD22drkz;
dD22drz dD22dz2 dD22dzkr dD22dzkz;
dD22drkr dD22dzkr dD22dkr2 dD22dkrkz;
dD22drkz dD22dzkz dD22dkrkz dD22dkz2 ];
% for the second order derivatives, we don't need dom
dD11_vec = [dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dr dD12dz dD12dkr dD12dkz];
dD22_vec = [dD22dr dD22dz dD22dkr dD22dkz];
dUdom = dU_vec(1);
dU_vec = dU_vec(2:end);
dU_mat = (1/subsum)*(...
dD11_mat*(D22-U) + dD22_mat*(D11-U)...
+(dD11_vec - dU_vec).'*(dD22_vec - dU_vec)...
+(dD22_vec - dU_vec).'*(dD11_vec - dU_vec)...
-2*real(cD12*dD12_mat + dD12_vec'*dD12_vec));
dU_vec = [dUdom dU_vec];
end
case {'cld3x3'}
% --- Elementary functions and derivatives ---------------------------------
if (isCmplx && ~eval1st)
iomceps=0.003*i*om2;
else
iomceps=0;
end
omc2Mom2= omc2-om2-2*iomceps; % S,D,P
Si=omp2./omc2Mom2; S=1+sum( Si ,2);
Di=(omc/plasma.omega).*Si; D= sum( Di ,2);
Pi=omp2/om2; P=1-sum( Pi ,2);
% 1st derivatives S,D,P
zroi=zeros(size(Si));
zro=zeros(size(S));
dLNSids =dLNnds-2.*omc2./omc2Mom2.*dLNomcds; dSds = sum( Si.*dLNSids ,2);
dLNSidt = -2.*omc2./omc2Mom2.*dLNomcdt; dSdt = sum( Si.*dLNSidt ,2);
dLNSidom =2*plasma.omega./omc2Mom2; dSdom= sum( Si.*dLNSidom,2);
dLNDids =dLNSids + dLNomcds; dDds = sum( Di.*dLNDids ,2);
dLNDidt =dLNSidt + dLNomcdt; dDdt = sum( Di.*dLNDidt ,2);
dLNDidom =(3*om2-omc2)./(plasma.omega*omc2Mom2); dDdom= sum( Di.*dLNDidom,2);
dLNPids = dLNnds;
dPds = -sum( Pi.*dLNPids ,2);
dLNPidt = zroi;
dPdt = zro;
dLNPidom = -2/plasma.omega;
dPdom = -sum( Pi.*dLNPidom,2);
% second derivatives
if eval2nd
omOM = om2 ./omc2Mom2;
ocOM = omc2./omc2Mom2;
dLNSids2 = 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcds -dLNomcds2)+dLNnds2;
dLNSidst = 2*ocOM.*(2*omOM.*dLNomcds.*dLNomcdt -dLNomcdst);
dLNSidt2 = 2*ocOM.*(2*omOM.*dLNomcdt.*dLNomcdt -dLNomcdt2);
dLNDids2 = dLNomcds2 +dLNSids2;
dLNDidst = dLNomcdst +dLNSidst;
dLNDidt2 = dLNomcdt2 +dLNSidt2;
dSids2 = Si.*(dLNSids2 +dLNSids.*dLNSids); dSds2 = sum( dSids2 ,2);
dSidst = Si.*(dLNSidst +dLNSids.*dLNSidt); dSdst = sum( dSidst ,2);
dSidt2 = Si.*(dLNSidt2 +dLNSidt.*dLNSidt); dSdt2 = sum( dSidt2 ,2);
dDids2 = Di.*(dLNDids2 +dLNDids.*dLNDids); dDds2 = sum( dDids2 ,2);
dDidst = Di.*(dLNDidst +dLNDids.*dLNDidt); dDdst = sum( dDidst ,2);
dDidt2 = Di.*(dLNDidt2 +dLNDidt.*dLNDidt); dDdt2 = sum( dDidt2 ,2);
dLNPids2 = dLNnds2; dLNPidst= zroi; dLNPidt2 = zroi;
dPids2 = Pi.*(dLNPids2 +dLNPids.*dLNPids); dPds2 = -sum( dPids2 ,2);
dPidst = Pi.*(dLNPidst +dLNPids.*dLNPidt); dPdst = -sum( dPidst ,2);
dPidt2 = Pi.*(dLNPidt2 +dLNPidt.*dLNPidt); dPdt2 = -sum( dPidt2 ,2);
end
% --- Tensor elements and derivatives --------------------------------------
D11 = Nb2+Np2-S;
D12 =-Nn.*Nb-D*i;
D13 =-Nn.*Np;
D22 = Nn2+Np2-S;
D23 =-Nb.*Np;
D33 = Nn2+Nb2-P;
cD12 = conj(D12);
cD13 = conj(D13);
cD23 = conj(D23);
% Dispersion tensor
DD=[D11 D12 D13 cD12 D22 D23 cD13 cD23 D33];
% We will assume that the input yv is size [1,4], then reshape works
DD=reshape(DD,3,3);
% Convert to double to make sure that double precision LAPACK routines
% are used.
[evects,evals]=eig(double(DD)); % Three values
evs = diag(evals);
ind = find(abs(evs)==min(abs(evs)));
U = evs(ind); % The eval nearest zero is the one we want
pol = evects(:,ind); % get the column with the right evector
% Conversion monitor
mon2 = abs(det(DD([2 3],[2 3])) + ...
det(DD([1 3],[1 3])) + ...
det(DD([1 2],[1 2])) );
if eval1st
dD11ds =-dSds; dD11dt =-dSdt;
dD12ds =-dDds*i; dD12dt =-dDdt*i;
dD13ds = zro; dD13dt = zro;
dD22ds =-dSds; dD22dt =-dSdt;
dD23ds = zro; dD23dt = zro;
dD33ds =-dPds; dD33dt =-dPdt;
dD11dkn= zro; dD11dkb= Nb*coom*2; dD11dkp= Np*coom*2;
dD12dkn=-Nb*coom; dD12dkb=-Nn*coom; dD12dkp= zro;
dD13dkn=-Np*coom; dD13dkb= zro; dD13dkp=-Nn*coom;
dD22dkn= Nn*coom*2; dD22dkb= zro; dD22dkp= Np*coom*2;
dD23dkn= zro; dD23dkb=-Np*coom; dD23dkp=-Nb*coom;
dD33dkn= Nn*coom*2; dD33dkb= Nb*coom*2; dD33dkp= zro;
dD11dkr=dD11dkn.*ener +dD11dkb.*eber +dD11dkp.*eper;
dD11dkz=dD11dkn.*enez +dD11dkb.*ebez +dD11dkp.*epez;
dD12dkr=dD12dkn.*ener +dD12dkb.*eber +dD12dkp.*eper;
dD12dkz=dD12dkn.*enez +dD12dkb.*ebez +dD12dkp.*epez;
dD13dkr=dD13dkn.*ener +dD13dkb.*eber +dD13dkp.*eper;
dD13dkz=dD13dkn.*enez +dD13dkb.*ebez +dD13dkp.*epez;
dD22dkr=dD22dkn.*ener +dD22dkb.*eber +dD22dkp.*eper;
dD22dkz=dD22dkn.*enez +dD22dkb.*ebez +dD22dkp.*epez;
dD23dkr=dD23dkn.*ener +dD23dkb.*eber +dD23dkp.*eper;
dD23dkz=dD23dkn.*enez +dD23dkb.*ebez +dD23dkp.*epez;
dD33dkr=dD33dkn.*ener +dD33dkb.*eber +dD33dkp.*eper;
dD33dkz=dD33dkn.*enez +dD33dkb.*ebez +dD33dkp.*epez;
dD11dom=-2/plasma.omega*(Nb2+Np2)-dSdom;
dD12dom=-2/plasma.omega*(Nn.*Nb) -dDdom*i;
dD13dom=-2/plasma.omega*(Nn.*Np);
dD22dom=-2/plasma.omega*(Nn2+Np2)-dSdom;
dD23dom=-2/plasma.omega*(Nb.*Np);
dD33dom=-2/plasma.omega*(Nn2+Nb2)-dPdom;
DD2=DD-eye(3)*U;
% Sub-blocks of (dispersion matrix - eigenval*id)
U1 = det(DD2([2 3],[2 3]));
U2 = det(DD2([1 3],[1 3]));
U3 = det(DD2([1 2],[1 2]));
subsum = U1+U2+U3;
% Convert to r, z coordinates.
% Note that there are extra correction terms due to field
% curvature. They are derivatives of the trasformation elements:
% (d/dr)(ebef) etc.
dNndr = Nr.*denerdr + Nf.*denefdr + Nz.*denezdr;
dNndz = Nr.*denerdz + Nf.*denefdz + Nz.*denezdz;
dNbdr = Nr.*deberdr + Nf.*debefdr + Nz.*debezdr;
dNbdz = Nr.*deberdz + Nf.*debefdz + Nz.*debezdz;
dNpdr = Nr.*deperdr + Nf.*depefdr + Nz.*depezdr;
dNpdz = Nr.*deperdz + Nf.*depefdz + Nz.*depezdz;
dD11dr = dD11ds.*dsdr + dD11dt.*dtdr + 2*(Nb.*dNbdr + Np.*dNpdr);
dD11dz = dD11ds.*dsdz + dD11dt.*dtdz + 2*(Nb.*dNbdz + Np.*dNpdz);
dD12dr = dD12ds.*dsdr + dD12dt.*dtdr - Nn.*dNbdr - Nb.*dNndr;
dD12dz = dD12ds.*dsdz + dD12dt.*dtdz - Nn.*dNbdz - Nb.*dNndz;
dD13dr = dD13ds.*dsdr + dD13dt.*dtdr - dNndr.*Np - Nn.*dNpdr;
dD13dz = dD13ds.*dsdz + dD13dt.*dtdz - dNndz.*Np - Nn.*dNpdz;
dD22dr = dD22ds.*dsdr + dD22dt.*dtdr + 2*(Nn.*dNndr + Np.*dNpdr);
dD22dz = dD22ds.*dsdz + dD22dt.*dtdz + 2*(Nn.*dNndz + Np.*dNpdz);
dD23dr = dD23ds.*dsdr + dD23dt.*dtdr - dNbdr.*Np - Nb.*dNpdr;
dD23dz = dD23ds.*dsdz + dD23dt.*dtdz - dNbdz.*Np - Nb.*dNpdz;
dD33dr = dD33ds.*dsdr + dD33dt.*dtdr + 2*(Nn.*dNndr + Nb.*dNbdr);
dD33dz = dD33ds.*dsdz + dD33dt.*dtdz + 2*(Nn.*dNndz + Nb.*dNbdz);
% Gather things together to do all derivatives at the same time
dD11_vec = [dD11dom dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dom dD12dr dD12dz dD12dkr dD12dkz];
dD13_vec = [dD13dom dD13dr dD13dz dD13dkr dD13dkz];
dD22_vec = [dD22dom dD22dr dD22dz dD22dkr dD22dkz];
dD23_vec = [dD23dom dD23dr dD23dz dD23dkr dD23dkz];
dD33_vec = [dD33dom dD33dr dD33dz dD33dkr dD33dkz];
% Derivatives of the eigenvalue
%dU_vec = [dUdom dUdr dUdz dUdkr dUdkz];
dU_vec = (dD11_vec.*U1 - (D11-U).*2.*real(cD23.*dD23_vec) ...
+dD22_vec.*U2 - (D22-U).*2.*real(cD13.*dD13_vec) ...
+dD33_vec.*U3 - (D33-U).*2.*real(cD12.*dD12_vec) )./subsum...
+ 2*real(D12.*D23.*conj(dD13_vec)...
+D12.*dD23_vec.*cD13...
+dD12_vec.*D23.*cD13)./subsum;
end
% terms needed for second derivative of hamiltonian
if eval2nd
dSdr2 = dsdr2.*dSds + dtdr2.*dSdt ...
+ dsdr.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdr.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdrz = dsdrz.*dSds + dtdrz.*dSdt ...
+ dsdz.*(dsdr.*dSds2 + dtdr.*dSdst) + dtdz.*(dsdr.*dSdst + dtdr.*dSdt2);
dSdz2 = dsdz2.*dSds + dtdz2.*dSdt ...
+ dsdz.*(dsdz.*dSds2 + dtdz.*dSdst) + dtdz.*(dsdz.*dSdst + dtdz.*dSdt2);
dDdr2 = dsdr2.*dDds + dtdr2.*dDdt ...
+ dsdr.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdr.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdrz = dsdrz.*dDds + dtdrz.*dDdt ...
+ dsdz.*(dsdr.*dDds2 + dtdr.*dDdst) + dtdz.*(dsdr.*dDdst + dtdr.*dDdt2);
dDdz2 = dsdz2.*dDds + dtdz2.*dDdt ...
+ dsdz.*(dsdz.*dDds2 + dtdz.*dDdst) + dtdz.*(dsdz.*dDdst + dtdz.*dDdt2);
dPdr2 = dsdr2.*dPds + dtdr2.*dPdt ...
+ dsdr.*(dsdr.*dPds2 + dtdr.*dPdst) + dtdr.*(dsdr.*dPdst + dtdr.*dPdt2);
dPdrz = dsdrz.*dPds + dtdrz.*dPdt ...
+ dsdz.*(dsdr.*dPds2 + dtdr.*dPdst) + dtdz.*(dsdr.*dPdst + dtdr.*dPdt2);
dPdz2 = dsdz2.*dPds + dtdz2.*dPdt ...
+ dsdz.*(dsdz.*dPds2 + dtdz.*dPdst) + dtdz.*(dsdz.*dPdst + dtdz.*dPdt2);
dNndr2 = Nr.*denerdr2 + Nf.*denefdr2 + Nz.*denezdr2;
dNndrz = Nr.*denerdrz + Nf.*denefdrz + Nz.*denezdrz;
dNndz2 = Nr.*denerdz2 + Nf.*denefdz2 + Nz.*denezdz2;
dNbdr2 = Nr.*deberdr2 + Nf.*debefdr2 + Nz.*debezdr2;
dNbdrz = Nr.*deberdrz + Nf.*debefdrz + Nz.*debezdrz;
dNbdz2 = Nr.*deberdz2 + Nf.*debefdz2 + Nz.*debezdz2;
dNpdr2 = Nr.*deperdr2 + Nf.*depefdr2 + Nz.*depezdr2;
dNpdrz = Nr.*deperdrz + Nf.*depefdrz + Nz.*depezdrz;
dNpdz2 = Nr.*deperdz2 + Nf.*depefdz2 + Nz.*depezdz2;
dD11dr2=2*(dNbdr.*dNbdr + Nb.*dNbdr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD11drz=2*(dNbdr.*dNbdz + Nb.*dNbdrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD11dz2=2*(dNbdz.*dNbdz + Nb.*dNbdz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD12dr2=-(Nn.*dNbdr2 + 2*dNndr.*dNbdr + Nb.*dNndr2)-i*dDdr2;
dD12drz=-(Nn.*dNbdrz + dNndr.*dNbdz + dNndz.*dNbdr + Nb.*dNndrz)-i*dDdrz;
dD12dz2=-(Nn.*dNbdz2 + 2*dNndz.*dNbdz + Nb.*dNndz2)-i*dDdz2;
dD13dr2=-(Nn.*dNpdr2 + 2*dNndr.*dNpdr + Np.*dNndr2);
dD13drz=-(Nn.*dNpdrz + dNndr.*dNpdz + dNndz.*dNpdr + Np.*dNndrz);
dD13dz2=-(Nn.*dNpdz2 + 2*dNndz.*dNpdz + Np.*dNndz2);
dD22dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNpdr.*dNpdr + Np.*dNpdr2) - dSdr2;
dD22drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNpdr.*dNpdz + Np.*dNpdrz) - dSdrz;
dD22dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNpdz.*dNpdz + Np.*dNpdz2) - dSdz2;
dD23dr2=-(Nb.*dNpdr2 + 2*dNbdr.*dNpdr + Np.*dNbdr2);
dD23drz=-(Nb.*dNpdrz + dNbdr.*dNpdz + dNbdz.*dNpdr + Np.*dNbdrz);
dD23dz2=-(Nb.*dNpdz2 + 2*dNbdz.*dNpdz + Np.*dNbdz2);
dD33dr2=2*(dNndr.*dNndr + Nn.*dNndr2 + dNbdr.*dNbdr + Nb.*dNbdr2) - dPdr2;
dD33drz=2*(dNndr.*dNndz + Nn.*dNndrz + dNbdr.*dNbdz + Nb.*dNbdrz) - dPdrz;
dD33dz2=2*(dNndz.*dNndz + Nn.*dNndz2 + dNbdz.*dNbdz + Nb.*dNbdz2) - dPdz2;
dD11drkr = 2*coom*(eber.*dNbdr + Nb.*deberdr + eper.*dNpdr + Np.*deperdr);
dD11drkz = 2*coom*(ebez.*dNbdr + Nb.*debezdr + epez.*dNpdr + Np.*depezdr);
dD11dzkr = 2*coom*(eber.*dNbdz + Nb.*deberdz + eper.*dNpdz + Np.*deperdz);
dD11dzkz = 2*coom*(ebez.*dNbdz + Nb.*debezdz + epez.*dNpdz + Np.*depezdz);
dD12drkr = -coom*(denerdr.*Nb + dNndr.*eber + ener.*dNbdr + Nn.*deberdr);
dD12drkz = -coom*(denezdr.*Nb + dNndr.*ebez + enez.*dNbdr + Nn.*debezdr);
dD12dzkr = -coom*(denerdz.*Nb + dNndz.*eber + ener.*dNbdz + Nn.*deberdz);
dD12dzkz = -coom*(denezdz.*Nb + dNndz.*ebez + enez.*dNbdz + Nn.*debezdz);
dD13drkr = -coom*(denerdr.*Np + dNndr.*eper + ener.*dNpdr + Nn.*deperdr);
dD13drkz = -coom*(denezdr.*Np + dNndr.*epez + enez.*dNpdr + Nn.*depezdr);
dD13dzkr = -coom*(denerdz.*Np + dNndz.*eper + ener.*dNpdz + Nn.*deperdz);
dD13dzkz = -coom*(denezdz.*Np + dNndz.*epez + enez.*dNpdz + Nn.*depezdz);
dD22drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eper.*dNpdr + Np.*deperdr);
dD22drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + epez.*dNpdr + Np.*depezdr);
dD22dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eper.*dNpdz + Np.*deperdz);
dD22dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + epez.*dNpdz + Np.*depezdz);
dD23drkr = -coom*(deperdr.*Nb + dNpdr.*eber + eper.*dNbdr + Np.*deberdr);
dD23drkz = -coom*(depezdr.*Nb + dNpdr.*ebez + epez.*dNbdr + Np.*debezdr);
dD23dzkr = -coom*(deperdz.*Nb + dNpdz.*eber + eper.*dNbdz + Np.*deberdz);
dD23dzkz = -coom*(depezdz.*Nb + dNpdz.*ebez + epez.*dNbdz + Np.*debezdz);
dD33drkr = 2*coom*(ener.*dNndr + Nn.*denerdr + eber.*dNbdr + Nb.*deberdr);
dD33drkz = 2*coom*(enez.*dNndr + Nn.*denezdr + ebez.*dNbdr + Nb.*debezdr);
dD33dzkr = 2*coom*(ener.*dNndz + Nn.*denerdz + eber.*dNbdz + Nb.*deberdz);
dD33dzkz = 2*coom*(enez.*dNndz + Nn.*denezdz + ebez.*dNbdz + Nb.*debezdz);
dD11dkr2 = 2*coomsq*(eber.^2 + eper.^2);
dD11dkz2 = 2*coomsq*(ebez.^2 + epez.^2);
dD11dkrkz= 2*coomsq*(eber.*ebez + eper.*epez);
dD12dkr2 = -coomsq*(ener.*eber + ener.*eber);
dD12dkz2 = -coomsq*(ener.*ebez + enez.*eber);
dD12dkrkz= -coomsq*(enez.*ebez + enez.*ebez);
dD13dkr2 = -coomsq*(ener.*eper + ener.*eper);
dD13dkz2 = -coomsq*(ener.*epez + enez.*eper);
dD13dkrkz= -coomsq*(enez.*epez + enez.*epez);
dD22dkr2 = 2*coomsq*(ener.^2 + eper.^2);
dD22dkz2 = 2*coomsq*(enez.^2 + epez.^2);
dD22dkrkz= 2*coomsq*(ener.*enez + eper.*epez);
dD23dkr2 = -coomsq*(eper.*eber + eper.*eber);
dD23dkz2 = -coomsq*(eper.*ebez + epez.*eber);
dD23dkrkz= -coomsq*(epez.*ebez + epez.*ebez);
dD33dkr2 = 2*coomsq*(ener.^2 + eber.^2);
dD33dkz2 = 2*coomsq*(enez.^2 + ebez.^2);
dD33dkrkz= 2*coomsq*(ener.*enez + eber.*ebez);
% dD11_mat = [ds dt dkr dkz].'*[ds dt dkr dkz];
dD11_mat = [dD11dr2 dD11drz dD11drkr dD11drkz;
dD11drz dD11dz2 dD11dzkr dD11dzkz;
dD11drkr dD11dzkr dD11dkr2 dD11dkrkz;
dD11drkz dD11dzkz dD11dkrkz dD11dkz2 ];
dD12_mat = [dD12dr2 dD12drz dD12drkr dD12drkz;
dD12drz dD12dz2 dD12dzkr dD12dzkz;
dD12drkr dD12dzkr dD12dkr2 dD12dkrkz;
dD12drkz dD12dzkz dD12dkrkz dD12dkz2 ];
dD13_mat = [dD13dr2 dD13drz dD13drkr dD13drkz;
dD13drz dD13dz2 dD13dzkr dD13dzkz;
dD13drkr dD13dzkr dD13dkr2 dD13dkrkz;
dD13drkz dD13dzkz dD13dkrkz dD13dkz2 ];
dD22_mat = [dD22dr2 dD22drz dD22drkr dD22drkz;
dD22drz dD22dz2 dD22dzkr dD22dzkz;
dD22drkr dD22dzkr dD22dkr2 dD22dkrkz;
dD22drkz dD22dzkz dD22dkrkz dD22dkz2 ];
dD23_mat = [dD23dr2 dD23drz dD23drkr dD23drkz;
dD23drz dD23dz2 dD23dzkr dD23dzkz;
dD23drkr dD23dzkr dD23dkr2 dD23dkrkz;
dD23drkz dD23dzkz dD23dkrkz dD23dkz2 ];
dD33_mat = [dD33dr2 dD33drz dD33drkr dD33drkz;
dD33drz dD33dz2 dD33dzkr dD33dzkz;
dD33drkr dD33dzkr dD33dkr2 dD33dkrkz;
dD33drkz dD33dzkz dD33dkrkz dD33dkz2 ];
% for the second order derivatives, we don't need dom
dD11_vec = [dD11dr dD11dz dD11dkr dD11dkz];
dD12_vec = [dD12dr dD12dz dD12dkr dD12dkz];
dD13_vec = [dD13dr dD13dz dD13dkr dD13dkz];
dD22_vec = [dD22dr dD22dz dD22dkr dD22dkz];
dD23_vec = [dD23dr dD23dz dD23dkr dD23dkz];
dD33_vec = [dD33dr dD33dz dD33dkr dD33dkz];
dUdom = dU_vec(1);
dU_vec = dU_vec(2:end);
dU1_vec = (dD22_vec - dU_vec).*(D33 - U)...
+ (dD33_vec - dU_vec).*(D22 - U)...
-2*real(cD23*dD23_vec);
dU2_vec = (dD11_vec - dU_vec).*(D33 - U)...
+ (dD33_vec - dU_vec).*(D11 - U)...
-2*real(cD13*dD13_vec);
dU3_vec = (dD22_vec - dU_vec).*(D11 - U)...
+ (dD11_vec - dU_vec).*(D22 - U)...
-2*real(cD12*dD12_vec);
dB_mat = 2*real( D23.*dD12_vec.'*conj(dD13_vec) ...
+ D12.*dD23_vec.'*conj(dD13_vec) ...
+ D12.*D23.*conj(dD13_mat) ...
+ dD12_vec.'*dD23_vec.*cD13 ...
+ D12.*dD23_mat.*cD13 ...
+ D12.*conj(dD13_vec).'*dD23_vec ...
+ dD12_mat.*D23.*cD13 ...
+ dD23_vec.'*dD12_vec.*cD13 ...
+ conj(dD13_vec).'*dD12_vec.*D23 );
dU_mat = (1/subsum).*( ...
dD11_mat*U1 + dD22_mat*U2 + dD33_mat*U3 ...
+dU1_vec.'*(dD11_vec-dU_vec) ...
+dU2_vec.'*(dD22_vec-dU_vec) ...
+dU3_vec.'*(dD33_vec-dU_vec) ...
-2*(dD11_vec-dU_vec).'*real(cD23*dD23_vec) ...
-2*(dD22_vec-dU_vec).'*real(cD13*dD13_vec) ...
-2*(dD33_vec-dU_vec).'*real(cD12*dD12_vec) ...
-(D11-U)*2*real(cD23*dD23_mat +dD23_vec'*dD23_vec) ...% ' not .' => conj
-(D22-U)*2*real(cD13*dD13_mat +dD13_vec'*dD13_vec) ...
-(D33-U)*2*real(cD12*dD12_mat +dD12_vec'*dD12_vec) ...
+dB_mat);
test = dU_mat;
if nnz( (test - test.') > 1E-10)
disp(test - test.');
error('disp_eig:cld3x3:symmetric','Matrix not symmetric');
end
dU_vec = [dUdom dU_vec];
end
% +++++ Warm plasma model, etc ++++++++++++++++++++++++++++++++++++++++++++++
otherwise
error(['Unknown model ''' plasma.MODEL(1:6) '''.']);
end;
%
% ===== Variable output dimensions ===========================================
%
switch oper
case {'Mon'} % --- Conversion monitors
mon1=0;
tmp=real(cat(1,mon1',mon2'));
out=reshape(tmp,1,numel(tmp));
case {'Dsp','Msw'} % --- Dispersion relations
out = real(U);
case {'Ten'} % --- Dispersion tensor
out = reshape(DD,9,1);
case {'Trj'} % --- ODE trajectory
% dUdom = dU_vec(1);
% dUdomM=(1./dUdom)'; % fn of t, not sigma
dUdomM = 1; % --- Solve as a fn of sigma, not t
J = [zeros(2,2) eye(2); -eye(2) zeros(2,2)]; % I think the sign here is different
% dU = [dUdr dUdz dUdkr dUdkz].';
dU = reshape(dU_vec(2:5),4,1);
dz = (J*dU);
if eval2nd
d2U = dU_mat;
ds = reshape(J*d2U*sym_mat,16,1);
out = real(dUdomM*([dz; ds ]));
else
out = real(dUdomM*dz);
end
% case {'Sgn'}
% out = sign(dUdom); % return the sign of the derivative of time wrt ray parameter
case {'Pol'}
out = pol;
otherwise
error(['Unknown oper ''' oper '''.']);
end