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cbspec.m
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%% Numerically integrates P_i(f,\theta)=\int dk^3 P_i(k) \delta(2\pi f - k*v)
%% for given kx(=2pi f/v, kx is in rotated frame with kx||v) and \theta
%% Model of P_i(k) is critically balanced with k_para~L^(1/3)k_perp^(2/3) in ion
%% inertial range and k_\para~L^(1/3)\rho^(1/3)k_\perp^(1/3) in electron inertial range
%% P_i(k) as described in Cho et al., 2002 (ApJ)
%%
%%
%% Input:
%% f Frequncy in Hz
%% theta Field-to-flow angle in degrees
%% n Number of nodes in one dimension
%% fun function to evaluate ( 1=exp, 2=expdamp, ... )
%% L Outer scale in m
%% rho Gyro radius (or else) in m
%% v Plasma bulk velocity in m/s
%% va Alfven velocity in m/s
%% B Magnetic field strength in nT
%% bounds Boundary in log10 for integration ( bounds = [kmin kmax] )
%% ratioTP Ratio between toroidal T and poloidal P fluctuations
%% rhoe Electron gyro radius (or else)
%% si_in Spectral index in [MHD kinetic] range
%% cb_in Critical balance exponent in [MHD kinetic] range
%%
%%
%% Author: Michael von Papen
%% Date: 26.06.2013
function [P,kern,Pkern] = cbspec (f,theta,fun,varargin)
%% Check Input
args=struct('B',5,...
'bounds',[-10 -2],...
'cb_in',[2/3 1/3],...
'isopol',0,...
'L',1e9,...
'mirror',0,...
'n',500,...
'ratioTP',0,...
'rho',1e5,...
'rhoe',NaN,...
'si_in',[-10/3 -11/3],...
'v',6e5,...
'va',6e4);
args=parseArgs(varargin,args);
if isnan(args.rhoe); args.rhoe=args.rho/42.85; end
L=args.L;
rho=args.rho;
rhoe=args.rhoe;
va=args.va;
%% Wave vector in rotated coordinate system
kix=2*pi*f/args.v;
%% Basic Parameter
si=args.si_in(1); %-10/3->k^{-5/3}
si2=args.si_in(2); %-11/3->k^{-7/3}
cb=args.cb_in(1); % 2/3->alfven
cb2=args.cb_in(2); %1/3->KAW
%% K-space gridpoints
% Set boundaries a little bit wider than kmin,kmax for numerical reasons.
% Later everything outside [kmin,kmax] will be disregarded
kmin=args.bounds(1);
kmax=args.bounds(2);
ky=10.^[kmin+(0:args.n-1)*(kmax-kmin)/args.n];
dky=[ky(2:end) 2*10^kmax]-ky;
%nz log verteilt auf pos UND neg Achse
kiz=repmat([-ky(end:-1:1) ky],args.n,1);
dkiz=repmat([dky(end:-1:1) dky],args.n,1);
ky=ky(ones(1,2*args.n),:)'; % <=> ky=repmat(ky',1,2*n);
dky=dky(ones(1,2*args.n),:)';
% %% Ion-cyclotron frequency cut-off
% % All k_para ~ w_ic/V_A are subject to ion-cyclotron damping
% % Thus, parallel scales cannot reach k_para >> w_ic/V_A
wic=1.6e-19*args.B*1e-9/1.67e-27; %wic for protons
%% Set output variable
if args.ratioTP ~=0
if args.isopol==1
P=zeros(6,length(kix),length(theta));
else
P=zeros(4,length(kix),length(theta));
end
else
P=zeros(length(kix),length(theta));
end
%% Begin with loop over theta
for k=1:length(theta)
% fprintf('theta=%3.1f°\n',theta(k))
%% Begin loop over frequency
for i=1:length(kix)
%% Calculate PSD at z
%% CAUTION: Setting kiz=0 on x-axis makes problems when determining
%% spectral anisotropy on MHD scales for small outer scales and/or
%% small angles
if theta(k)==0
kx=kix(i)*sind(theta(k))-kiz.*cosd(theta(k)); %=kx in unrotated system
%ky=kiy in unrotated system
kz=kix(i)*cosd(theta(k))+kiz.*sind(theta(k)); %=kz in unrotated system
else
kx=kix(i)*sind(theta(k))-kiz.*cosd(theta(k))+kix(i).*cosd(theta(k))^2/sind(theta(k)); %=kx in unrotated system with kiz=0 on x-axis
%ky=kiy in unrotated system
kz=kiz.*sind(theta(k)); %=kz in unrotated system with kiz=0 on x-axis
end
kern=zeros(args.n,2*args.n);
kp2=ky.^2+kx.^2; % k_perp^2
kabs2=kp2+kz.^2; % |k|
% %% Get damping factors from Anne, Tmax
% load('/afs/geo/usr/vonpapen/PhD/publication/cbmodel/anne_data/anne.mat')
% efi=interp1(kp,damp_factor_chen',sqrt(kp2));
%% Equations written in unprimed coordinates for the sake of
%% brevity, but integration is done over primed variables,
%% which is why dkiy and dkiz is used.
%% Single components Alfven cascade
% take out '& kabs2 > 1/L^2' when checking for fmax or showing L-dependence
i1=find(kabs2 > 1/L^2 & kp2 <= 1/rho^2);
switch fun
case 1 %'exp'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-L^(1-cb).*abs(kz(i1))./kp2(i1).^(cb/2) )...
.*dky(i1).*dkiz(i1);
case 2 %'expdamp'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-L^(1-cb).*abs(kz(i1))./kp2(i1).^(cb/2)...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 3 %'gauss'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-(L^(1-cb)*abs(kz(i1))./kp2(i1).^(cb/2)-1).^2)...
.*dky(i1).*dkiz(i1)/sqrt(pi);
case 4 %'gaussdamp'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-(L^(1-cb)*abs(kz(i1))./kp2(i1).^(cb/2)-1).^2 ...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1)/sqrt(pi);
case 5 %'heavi'
i2=find(L^(1-cb)*abs(kz(i1)./kp2(i1).^(cb/2))<=1);
kern( i1(i2) ) = kp2(i1(i2)).^(si/2).*dky(i1(i2)).*dkiz(i1(i2));
case 6 %'delta'
[tmp,i2]=min((L^(1-cb)*abs(kz(i1))-kp2(i1).^(cb/2)).^2);
kern( i1(i2) ) = L^(1-cb).*kp2(i1(i2)).^(si/2)...
.*dky(i1(i2)).*dkiz(i1(i2));
case 7 %'expisodamp'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 8 %'exp anne damp'
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-L^(1-cb).*abs(kz(i1))./kp2(i1).^(cb/2)-abs(kz(i1))*va/wic)...
.*dky(i1).*dkiz(i1).*efi(i1);
case 9 %'exp' damped only by IC damping
kern( i1 ) = kp2(i1).^(si/2)...
.*exp(-L^(1-cb).*abs(kz(i1))./kp2(i1).^(cb/2)...
-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
end
%% Single components KAW cascade
i1=find(kabs2 > 1/L^2 & kp2 > 1/rho^2 & kp2 <= 1/rhoe^2);
switch fun
case 1 %'exp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2) )...
.*dky(i1).*dkiz(i1);
case 2 %'expdamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 3 %'gauss'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)-1).^2)...
.*dky(i1).*dkiz(i1)/sqrt(pi);
case 4 %'gaussdamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)-1).^2 ...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1)/sqrt(pi);
case 5 %'heavi'
i2=find(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)<=1);
kern( i1(i2) ) = rho^(si2-si).*kp2(i1(i2)).^(si2/2)...
.*dky(i1(i2)).*dkiz(i1(i2));
case 6 %'delta'
[tmp,i2]=min((L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))-kp2(i1).^(cb2/2)).^2);
kern( i1(i2) ) = L^(1-cb)*rho^(si2-si).*kp2(i1(i2)).^(si2/2)...
.*dky(i1(i2)).*dkiz(i1(i2));
case 7 %'expisodamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp(-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 8 %'exp anne damp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)-abs(kz(i1))*va/wic )...
.*dky(i1).*dkiz(i1).*efi(i1);
case 9 %'exp' damped only by IC damping
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))./kp2(i1).^(cb2/2)...
-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
end
%% Single components of cascade at electron scales
i1=find(kabs2 > 1/L^2 & kp2 > 1/rhoe^2);
switch fun
case 1 %'exp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2 )...
.*dky(i1).*dkiz(i1);
case 2 %'expdamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 3 %'gauss'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2-1).^2)...
.*dky(i1).*dkiz(i1)/sqrt(pi);
case 4 %'gaussdamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2-1).^2 ...
-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1)/sqrt(pi);
case 5 %'heavi'
i2=find(L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2<=1);
kern( i1(i2) ) = rho^(si2-si).*kp2(i1(i2)).^(si2/2)...
.*dky(i1(i2)).*dkiz(i1(i2));
case 6 %'delta'
[tmp,i2]=min((L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))-rhoe^cb2).^2);
kern( i1(i2) ) = L^(1-cb)*rho^(si2-si).*kp2(i1(i2)).^(si/2)...
.*dky(i1(i2)).*dkiz(i1(i2));
case 7 %'expisodamp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp(-sqrt(kp2(i1))*rhoe-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
case 8 %'exp'
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2-abs(kz(i1))*va/wic )...
.*dky(i1).*dkiz(i1).*efi(i1);
case 9 %'exp' damped only by IC damping
kern( i1 ) = rho^(si2-si).*kp2(i1).^(si2/2)...
.*exp( -L^(1-cb)*rho^(cb-cb2)*abs(kz(i1))*rhoe^cb2...
-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
end
%% Full version with Toroidal and Poloidal parts
% Sum up to get power for one ky value
if args.ratioTP ~= 0
Tor=kern./kp2;
% Add mirror modes to Pkern %WIP
if args.mirror~=0
sigma=0.2; sigmaz=1e-6;
i1=find(kz.^2./kp2<1e-2); %populate Pkern up to certain kz
% Energy distribution of mirror mode modelled as gaussian
% in logspace for kp*rho and normal gaussian for kz
kern(i1)=kern(i1)+args.mirror/(2*pi*sqrt(sigma*sigmaz))...
*exp(-log10(sqrt(kp2(i1))*rho).^2/2/sigma^2 ...
-kz(i1).^2/2/sigmaz^2).*dky(i1).*dkiz(i1);
end
%% Compose Poloidal Spectrum
if args.isopol==1
i1=find(kp2 > 1/L^2 & kp2 <= 1/rhoe^2 & kabs2 <= 10^(2*kmax));
Pkern=zeros(args.n,2*args.n);
Pkern( i1 ) = kabs2(i1).^(-2)/L^(2/3) ...
.*exp(-sqrt(kp2(i1))*rho-abs(kz(i1))*va/wic).*dky(i1).*dkiz(i1);
Pol=(Pkern+kern)/args.ratioTP./kabs2;
else
Pol=kern/args.ratioTP./kabs2;
end
%% Add Spectra to PSD
P(1,i,k)=sum(sum( ky.^2.*Tor ...
+(kx.*kz).^2./kp2.*Pol ));
P(2,i,k)=sum(sum( kx.^2.*Tor ...
+(ky.*kz).^2./kp2.*Pol ));
P(3,i,k)=sum(sum( kp2.*Pol ));
P(4,i,k)=sum(P(1:3,i,k));
% if isopol==1
% P(5,i,k)=sum(sum( (1-ratioTP/(1+ratioTP))*kp2.*kern./kabs2 ));
% P(6,i,k)=sum(sum( (1-ratioTP/(1+ratioTP))*kp2.*Pkern./kabs2 ));
% end
else
P(i,k)=sum(sum(kern));
end
end
end
%% Multiply Spectra with factor to obtain PSD in nT^2/Hz
P=2*args.B^2/args.v/L^(1-cb).*P;