Baroklina nestabilnost Navodila za projektno nalogo iz dinamične meteorologije 2012/2013 Januar 2013 Nedjeljka Zagar in Rahela Zabkar Naloga je zasnovana na dvoslojnem modelu baroklinega razvoja, napisana je v Matlabu. Naloga je prevzeta iz Oddelka za meteorologijo na Univerzi v Stockholmu, kjer je ena od nalog pri predmetu Dinamična meteorologija. Avtorji originalne kode in navodil so Erland Källén, Gerhard Erbes, Shuting Yang, Robert Sigg in Rezwan Mohammad.
1 Theory In these applied exercises we will examine the properties of a simple model which can describe baroclinic instability. We will start from the two-layer model described by Holton (Holton, 2004, section 8.2). We will add dissipation terms. The equations are U1 and U3 describe the zonal background state and, ψ 3, and ω 2 the perturbation field. The dissipation terms are given with the coefficients ε i, ε D, and δ. The terms with the coefficient ε i describe exchange of kinetic energy between the two layers due to dissipation processes. In the atmosphere this corresponds to convective processes which are not explicitly described by this simple model. The dimension of ε i is [s -1 ] (the dissipation works directly on the vorticity). The characteristic dissipation timescale in the atmosphere is about ten days. The dissipation coefficient ε D represents the friction against the surface and has a characteristic time-scale about five days. The dissipation in the thermodynamic equation corresponds to a cooling of the middle layer which has effect on a time-scale of ten days. The dissipation terms are simple parameterizations of complicated processes in the atmosphere. Hence the values of the dissipation coefficients are a rough estimate. One of the aims of these exercises is to study the sensitivity of this model to variations in these parameters. To simplify the analysis we assume that the dissipation terms are proportional to each other. Hence we assign the parameter ε as follows: ε = ε D = 2ε i = 2δ Now ε represents all the dissipation processes. We can reduce the equation system (equation 1) by eliminating the vertical velocity ω 2.
where and D 1 and D 2 are the dissipation terms which you will derive in exercise1. Del I: Theoretical exercises Exercise 1 Derive the dissipation terms D 1 and D 2 in equation system 2. Exercise 2 If we assume wave-like solutions of the form the equation system 2 can be written in matrix form as follows: (3) Determine the matrix elements in equation 3. The stability of this linear system can be examined by determine the eigen values of the matrix M in equation 3. If any eigen value has a positive real part, the solution is unstable. Exercise 3 Derive the relation between the static stability parameter σ and the Brunt-Väisälä frequency N. Exercise 4 Try to describe physically why the presence of the boundary layer restrains baroclinic instabilities. Use Holton (2004, section 5.4, or 1992,) or any other source which discusses the secondary circulation and spin down within the planetary boundary layer.
Del II: Matlab exercises (Katalog Matlab vsebuje 5 programov ki računajo in rišejo rezultate dvoslojnega modela) Matlab program eigenplot.m calculates the eigen values to the matrix in equation 3 for given parameter values. For given parameter values, program calculates the eigen values. In addition it examines for which value of U T that instability occurs. Input parameters are non-dimensional wavenumber k * dissipation time scale ε -1 (in seconds) Brunt-Väisälä frequency N Latitude ϕ o (degrees) Wind speed at level 3 U 3 The non-dimensional wavenumber k * must be an integer. It represents the number of waves around one latitude circle at the latitude ϕ o. Hence the non-dimensional wave number is given by k * = k R a cos(ϕ o ), where k is the actual wave number [m -1 ], R a is the radius of the earth (6:37 10 6 m) and ϕ o is the central latitude. Exercise 1: No dissipation >> k_nondim = 6; >> dissip = 100*86400; % (100 dni) Velika vrednost pomeni, da ni disipacije (druga opcija je dissip = Inf;) >> latitude = 45; >> U_3 = 0; % Plot eigen values and phase speed and calculate minimum wind shear >> figure(1) >> U_T_min = eigenplot(k_nondim, dissip, N, latitude, U_3); % Run loop [k_nondim, U_T_min] = eigenloop(dissip, N, latitude, U_3); % Plot the neutral instabillity graph figure(2) marginal(dissip, N, latitude, U_m) a) Set the dissipation coefficients to very small values and verify Figure 8.3 in Holton (krivulja neutralne stabilnosti za dvoslojni baroklini model) b) Study how the eigen values vary as a function of U 1 (U 3 = 0), for one wavenumber just below and for one just above the cut-off wavenumber. Exercise 2: With dissipation >> dissip = 10*86400; >> latitude = 45;
% Mean wind speed >> U_m = 10; % Plot the neutral instabillity graph figure(2) marginal(dissip, N, latitude, U_m). % Plot the neutral instabillity graph for various dissipations >> dissip = [1 1.2 2 5 10 Inf]*86400; >> [k_nondim, U_T_min] = marginal(dissip, N, latitude, U_m); >> Handle = legend(num2str(dissip'/86400), -1); >> set(get(handle, 'XLabel'), 'String', 'Dissipation') >> title('neutral instability graph') >> xlabel('non-dimensional wave number') >> ylabel('wind shear [m/s]') % Plot the horizontal structure for ψ m and ψ T >> U_T = 20; >> k_nondim = 6; >> dissip = 86400; >> latitude = 45; >> phase_shift = structure(u_t, k_nondim, dissip, N, latitude); % Plot the neutral instabillity graph for various latitudes >> dissip = Inf; >> latitude = [30:10:80]; >> U_m = 10; >> [k_nondim, U_T_min] = marginal(dissip, N, latitude, U_m); >> Handle = legend(num2str(latitude'), -1); >> set(get(handle, 'XLabel'), 'String', 'Latitude') >> title('neutral instability graph') >> xlabel('non-dimensional wave number') >> ylabel('wind shear [m/s]') a) Set the dissipation coefficients to a plausible value (see the theory section). Study the sensitivity of the instability for variations of the dissipation. b) Study if the mean wind speed U m has influence on the instability. c) What are your conclusions of the sensitivity study? d) Give a physical explanation of the stabilization of the long respectively the short waves.
Finally Play around with the programs to investigate the effect of other choices of parameter values. For instance, vary the Brunt-Väisälä frequency and the latitude. The physical motivation of the choice of a parameter value should be clear in the laboration report. References James R. Holton. An introduction to dynamic meteorology, volume 48 of International geophysics series. Academic Press, Inc., third edition, 1992., fourth edition, 2004. Nalogo je potrebno oddati do 1. marca 2013.