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⛄ 内容介绍
大气边界层中 Ekman 方程解的 Matlab 实现。假设流动是水平和均匀的。然而，可以模拟高度相关的涡流粘度。解决方案以一维形式提供。
EkmanAnalytic 函数为大气边界层中的恒定涡流粘度提供埃克曼方程的分析解。函数 solveEkman 用显式有限差分格式对 Ekman 方程进行数值求解，并允许使用高度相关的涡粘性。数值实现部分受到 [1] 的启发。
⛄ 部分代码
%% InputParser
p = inputParser();
p.CaseSensitive = false;
p.addOptional('Nmax',2e4);
p.addOptional('dt',[]);
p.addOptional('DT_save',100);
p.addOptional('Omega',7.29e-5);
p.addOptional('critConv',1e-6);
p.addOptional('method','Euler');
p.parse(varargin{:});
%%%%%%%%%%%%%%%%%%%%%%%%%%
Nmax = p.Results.Nmax ; % Maximal number of time steps (for convergence)
dt = p.Results.dt ; % time step for RK4
Omega = p.Results.Omega; % rate of angular rotation of the Earth (rad/s)
critConv= p.Results.critConv; % convergence criterion
DT_save= p.Results.DT_save; % Intermediate value are stored every DT step
method= p.Results.method;
%% Preallocation and Initial conditions
Nz = numel(z);
f = 2*Omega*sind(latitude);
dz = [0,diff(z)];
ut = zeros(Nmax,Nz);
vt = zeros(Nmax,Nz);
e = ones(Nz, 1);
A = spdiags( [e -2*e e], -1:1, Nz, Nz);
if isempty(dt)
dt = 0.4*dz.^2/max(K);
end
if (dt>0.48*dz.^2/max(K)), warning('Time step may be too large.'); end
%% Initial conditions
u = Ug.*ones(1,Nz);
v = zeros(1,Nz);
%% Main loop
myFun = @(Y,A,F) A*Y + F;
count= 1;
for time_step=1:Nmax % time step
% update u,v a nd time at each step
oldU = u;
oldV = v;
if strcmpi(method,'Euler')
[u,v] = Euler(u,v,dt,Ug,K,f,A,dz);
elseif strcmpi(method,'RK4')
[u,v] = solve_with_RK4(myFun,u,v,Ug,K,f,A,dt,dz);
end
% Boundary conditions
[u,v] = applyBC(u,v,Ug,Nz);
% Store data very N step
if mod(time_step,DT_save)==0
ut(count,:)=u;
vt(count,:)=v;
count = count+1;
if time_step>1 && time_step<Nmax
% Check convergence
maxDiffU = max(abs(u-oldU));
maxDiffV = max(abs(v-oldV));
if maxDiffU < critConv && maxDiffV< critConv
ut = ut(1:count-1,:);
vt = vt(1:count-1,:);
t = [0:count-2].*mean(dt);
fprintf('The algorithm has converged \n')
return
end
elseif time_step==Nmax
ut = ut(1:count-1,:);
vt = vt(1:count-1,:);
t = [0:count-2].*mean(dt);
warning('The algorithm did not converge')
else
error('Unknown error')
end
end
end
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function [u,v] = Euler(u,v,dt,Ug,K,f,A,dz)
% At each time step, compute the second derivative of u and v
d2u = (A*u')'./dz.^2.*K;
d2v = (A*v')'./dz.^2.*K;
if numel(K)==1
u = u + dt.*(d2u + f.*v) ;
v = v + dt.*(d2v + f.*(Ug-u));
else
dK = [0,diff(K)];
dU = [0,diff(u)];
dV = [0,diff(v)];
u = u + dt.*(d2u + f.*v + dK.*dU./dz.^2);
v = v + dt.*(d2v + f.*(Ug-u)+ dK.*dV./dz.^2);
end
end
function [u,v] = applyBC(u,v,Ug,Nz)
% apply the upper boundary condition
u(Nz) = Ug;
v(Nz) = 0;
% apply the no-slip lower boundary condition
u(1) = 0;
v(1) = 0;
end
function [u,v] = solve_with_RK4(Fun,u,v,Ug,K,f,A,dt,dz)
% dt = repmat(w*(dz.^2)./K,2,1); % time step
% At each time step, compute the second derivative of u and v
d2u = K.*(A*u')'./dz.^2;
d2v = K.*(A*v')'./dz.^2;
if numel(K)==1
F = [d2u; d2v + f.*Ug];
else
dK = [0,diff(K)];
dU = [0,diff(u)];
dV = [0,diff(v)];
F = [d2u + dK.*dU./dz.^2; d2v + dK.*dV./dz.^2 + f.*Ug];
end
Y = [u;v];
B = [0 f; -f 0];
% Runge-Kutta of order 4
k_1 = Fun(Y,B,F);
k_2 = Fun(Y+0.5*dt.*k_1,B,F);
k_3 = Fun(Y+0.5*dt.*k_2,B,F);
k_4 = Fun(Y+k_3.*dt,B,F);
% output
Y = Y + (1/6)*(k_1+2*k_2+2*k_3+k_4).*dt;
u = Y(1,:);
v = Y(2,:);
% dt = dt(1,:);
end
end
⛄ 运行结果
⛄ 参考文献
[1] Berger, BW, & Grisogono, B. (1998)。斜压变涡粘性埃克曼层。边界层气象学，87（3），363-380。
[2] https://www.whoi.edu/science/PO/people/jprice/website/education_scripts.html
[3]伍荣生. 地形与Ekman边界层中的气流[J]. 气象学报, 1989, 47(2):10.
[4]宗金星. 非定常,水平非均匀的正压Ekman边界层的动力特征[J]. 气象科学, 1990, 10(3):14.
[5]刘敏. Ekman层风随高度分布方程的数值解法[C]// 中国地球物理学会第二十届年会论文集. 2004.
