Evaluate the theoretical vertical profiles (or Bottom Ekman spiral profiles)
of tidal currents in the two rotary directions driven by half-unit of sea
surface gradients in the two directions respectively. Eddy viscosity is
assumed as vertically invariant. See tidal_ellipse.ps for more details.
inputs:
g, the gravity acceleration,
f, the Coriolis parameter,
nu, the eddy viscosity
kappa, the bottom frictional coefficient
z, the vertical coordinates, can be a vector but must be
within [0 -h];
h, the water depth, must be positive.
Note: except for z, all other inputs must be scalars.
outputs:
BEp and BEm, the same dimensions of z, the outputs for the vertical
velocity profiles driven respectively by a unit of sea
surface slope in the positive rotation direction and negative
rotation direction for when the eddy viscosity is vertically
invariant. See the associated document for more details.0001 function [BEp, BEm]=cBEpm(g, f, sigma, nu, kappa, z, h) 0002 % 0003 %Evaluate the theoretical vertical profiles (or Bottom Ekman spiral profiles) 0004 %of tidal currents in the two rotary directions driven by half-unit of sea 0005 %surface gradients in the two directions respectively. Eddy viscosity is 0006 %assumed as vertically invariant. See tidal_ellipse.ps for more details. 0007 % 0008 % 0009 %inputs: 0010 % 0011 % g, the gravity acceleration, 0012 % f, the Coriolis parameter, 0013 % nu, the eddy viscosity 0014 % kappa, the bottom frictional coefficient 0015 % z, the vertical coordinates, can be a vector but must be 0016 % within [0 -h]; 0017 % h, the water depth, must be positive. 0018 % 0019 % Note: except for z, all other inputs must be scalars. 0020 % 0021 %outputs: 0022 % 0023 % BEp and BEm, the same dimensions of z, the outputs for the vertical 0024 % velocity profiles driven respectively by a unit of sea 0025 % surface slope in the positive rotation direction and negative 0026 % rotation direction for when the eddy viscosity is vertically 0027 % invariant. See the associated document for more details. 0028 0029 if length(g)>1 | length(f)>1 | length(sigma)>1 | ... 0030 length(nu)>1 | length(kappa)>1 | length(h)>1 0031 error('inputs of g,f,sigma, nu, kappa, and h should be all scalars!') 0032 end 0033 0034 if any(z/h>0) | any(z/h<-1) 0035 disp('z must be negative and must be within [0 -h]') 0036 end 0037 0038 delta_e = sqrt(2*nu/f); %Ekman depth 0039 alpha = (1+i)/delta_e*sqrt(1+sigma/f); 0040 beta = (1+i)/delta_e*sqrt(1-sigma/f); 0041 0042 BEp = get_BE(g, alpha, h, z, nu, kappa); 0043 BEm = get_BE(g, beta, h, z, nu, kappa); 0044 0045 0046 %subfunction 0047 %______________________________________________ 0048 function BE=get_BE(g, alpha, h, z, nu, kappa) 0049 0050 z = z(:); 0051 z_h = z/h; 0052 ah = alpha*h; 0053 az = alpha*z; 0054 ah2 = ah*2; 0055 anu_k = alpha*nu/kappa; 0056 nu_kh = nu/(kappa*h); 0057 0058 if abs(ah) < 1 %series solution 0059 T = 10; 0060 C = -g*h*h/(nu*(1+anu_k*tanh_v5_2(ah)))*2; 0061 A1 = (1-z_h.*z_h)/2+nu_kh; 0062 B1 = exp(-ah)/(1+exp(-ah2)); 0063 B = B1; 0064 series_sum=A1*B1; 0065 0066 for t = 2:T; 0067 t2=2*t; 0068 A = (1-z_h.^t2)./t2+nu_kh; 0069 B = B*ah*ah/(t2-1)/(t2-2); 0070 series_sum = series_sum+A*B; 0071 end 0072 0073 BE = C*series_sum; 0074 0075 else %finite solution 0076 0077 c = -g*h*h/nu; 0078 denom=(exp(az-ah)+exp(-(az+ah)))./(1+exp(-2*ah)); 0079 % =cosh(az)/cosh(ah); 0080 %but this a better way to evaluate it. 0081 0082 numer=1+anu_k*tanh_v5_2(ah); 0083 % BE=c*(1-denom/numer); 0084 BE = c*((1-denom/numer)/(ah*ah)); 0085 end 0086 0087 %end of subfunction 0088 % 0089 %Note tanh_v5_2 is a copy of tanh from Matlab v5.2, which has worked well! 0090 %It seems that Matlab v5.3 has some bug(s) in tanh function! It cannot deal 0091 %with large argument. try z=7.7249e02*(1+i), tanh(z) and tanh_v5_2(z) to 0092 %see the difference. 0093 0094 %Authorship Copyright: 0095 % 0096 % The author of this program retains the copyright of this program, while 0097 % you are welcome to use and distribute this program as long as you credit 0098 % the author properly and respect the program name itself. Particularly, 0099 % you are expected to retain the original author's name in this original 0100 % version of the program or any of its modified version that you might make. 0101 % You are also expected not to essentially change the name of the programs 0102 % except for adding possible extension for your own version you might create, 0103 % e.g. ap2ep_xx is acceptable. Any suggestions are welcome and enjoying my 0104 % program(s)! 0105 % 0106 % 0107 %Author Info: 0108 %_______________________________________________________________________ 0109 % Zhigang Xu, Ph.D. 0110 % (pronounced as Tsi Gahng Hsu) 0111 % Research Scientist 0112 % Coastal Circulation 0113 % Bedford Institute of Oceanography 0114 % 1 Challenge Dr. 0115 % P.O. Box 1006 Phone (902) 426-2307 (o) 0116 % Dartmouth, Nova Scotia Fax (902) 426-7827 0117 % CANADA B2Y 4A2 email zhigangx@emerald.bio.dfo.ca 0118 % zhigang_xu_98@yahoo.com 0119 %_______________________________________________________________________ 0120 % 0121 %Release Date: Nov. 2000 0122