MIKE 21 – Spectral Wave Model FM MIKE 21 – 谱波浪模型

1. MIKE 21 – Spectral Wave Model FM MIKE 21 – 谱波浪模型 Training Course Scientific Description Ole Svenstrup Petersen Head of Innovations Coastal & Estuarine Engineering E-mail: osp@dhigroup.com

2. Model features 模型特征 • Fully spectral (FS) and directionally decoupled parametric (DS) formulations • 全谱（FS）和方向解耦参数（DS）方程 • Instationary and quasi-stationary formulations • 非定常和准定常公式 • Source functions based on state-of-the-art formulations • 基于最新公式的源函数 • Optimal degree of flexibility in describing problems of different characteristic scales and bathymetry and ambient flow conditions using unstructured depth-adaptive and boundary-fitted mesh • 采用适于水深和拟合边界的非结构网格描述不同问题的灵活性 • Dynamic coupling with hydrodynamic flow model for modeling of wave-current interaction and time-varying water depth • 与水动力模型动态耦合，模拟水流相互作用和时间变化的水深条件

3. Basic variables and definitions 基本变量和定义 • Independent variables • 独立变量 • Cartesian coordinates • x-coordinate : x [m] • y-coordinate : y [m] • Spherical coordinates • latitude 纬度 :  [radian] • longitude 经度 :  [radian] • Time 时间 : t [s] • Absolute angular frequency绝对角频率 :  = 2fa[1/s] • Relative angular frequency 相对角频率 :  = 2f [1/s] • Wave direction波向: [radian] • Wave number vector 波数 : [1/m]

4. Basic variables and definitions 基本变量和定义 Linear dispersion relation线性耗散关系 where g is the acceleration of gravity, d is the water depth and is the current velocity vector. g是重力加速度，d是水深，是水流速度矢量 Phase velocity 相位速度 Group velocity

5. Basic variables (FS) 基本变量（FS） • Dependent variables (cartesian coordinates) • 独立变量（笛卡尔坐标系） • Energy density: [m2s/radian] • 能量密度 • Action density:[m2s2/radian] • 作用密度 • Relation between E and N • E和N的关系

6. Basic variables (FS) • Dependent variables (spherical coordinates) • 独立变量（球坐标系） • Energy density: [m4s/radian] • Action density: [m4s2/radian] • Relation between N and

7. Basic variables (FS) • Derived variables (integral parameters) • 推导变量（积分参数） • Spectral moments 波谱 • Basic wave parameters基本波浪参数 • Significant wave height:有效波高 • Mean period:平均周期 • Energy averaged mean period: • 能量平均周期 • Zero-crossing period:跨零周期

8. Governing equations (FS) • Wave action balance equation波作用守恒方程 • where is the action density, is the angular frequency, is the • direction of wave propagation, is the propagation velocity and • is the source term for the energy balance equation (rate at which E is generated • and dissipated) • 其中是作用密度，σ是角频率，θ是波向，是波群在四个方向上的传播速度，S是波能平衡方程的源项（当E在产生和消散时的速率。

9. Governing equations (FS) • Propagation velocity 传播速度 • Source terms 源项

10. Governing equations (FS) A deterministic prognostic part Wave action balance equation in the range from to An analytical diagnostic part where is the Pierson-Moskowitz peak frequency and is the mean frequency

11. Governing equations (FS) Seperation of wind-sea and swell 风浪和涌浪分离 Swell is defined as where the threshold frequency is constant or defined as For a fully developed wind-sea described by Pierson-Moskowitz spectrum (Young, 1999)

12. Basic variables (DS) • Dependent variables 独立变量 • Energy density: [m2s/radian] • Action density:[m2s2/radian] • Zero-th and first-order moment of the action density in the • frequency domain 在频率范围的波作用密度谱的零阶和一阶矩 • Wave functions波浪函数 • directional action spectrum • 方向作用谱 • action-averaged frequency • 平均作用频率

13. Basic variables (DS) • Additionals variables 其他变量 • Wave fucntions • directional energy density • 方向能量密度 • energy-averaged frequency • 能量平均频率 • Shape factors • 形状因子 • JONSWAP spectrum: • C1=0.92 and C2=0.83

14. Governing equations (DS) • Conservation equations for m0(θ) and m1(θ) • m0(θ)和m1(θ)的守恒方程 • Propagation velocity • 传播速度 • m0(θ) and m1(θ) is assumed to propagate with • identical propagation speeds which depend on ω0(θ)

15. Governing equations (DS) • Source terms 源项 • where SE and SΩ, respectively, are the rate at which E0(θ) and Ω0(θ) • are generated and dissipatedand where • S = Swind+ Sbot + Ssurf

16. Source terms (FS) • Wind • The wind input, Sin, is based on Janssens’s quasi-linear theory of wind wave generation (Janssen,1989, 1991), and implemented as • in WAM Cycle 4 (see Komen et al., 1994). • 风的输入项Sin,基于Janssen (1989), Janssen et al. (1989) and Janssen (1991) 的研究，作为WAM Cycle 4 (see Komen et al., 1994).的补充。

17. Source terms (FS) • Energy transfer能量传递 • Quadruplet-wave (four-wave) interaction四波相互作用 • The non-linear energy transfer through the four-wave interaction • is represented by the discrete interaction approximation (DIA) • proposed by Hasselmann et al. (1985) • 谱波浪模型中的四波相互作用基于公认的近似离散作用方法（DIA）。 • The quadruplet-wave interaction controls • 四波相互作用控制以下几个方面 • the shape-stabilisation of the high-frequency part of the spectrum谱的高频部分的形态稳定 • the downshift of energy to lower frequencies能量从高频区向低频区的转移 • frequency-dependent redistribution of directional distribution functions方向分布函数的基于频率的重新分布

18. Source terms (FS) • Energy transfer • Triad-wave interaction三波相互作用 • The triad-wave interaction is modeled using the simplified approach proposed by Eldeberky and Battjes (1995, 1996). • 对三波相互作用采用简化的方法（Eldeberky和Battjes，1995、1996）模拟 • The triad-wave interaction controls • 三波相互作用控制 • Generation of sub- and super harmonics – only super-harmonics are included in the formulation by Eldeberky and Battjes • 子谐波和超谐波的产生

19. Source terms (FS) • Dingemans bar test (Regular waves: T= 2s, H= 0.02m)

20. Source terms (FS) • LIP experiments

21. Source terms (FS) • Whitecapping白帽 • The source function describing the dissipation due to whitecapping, Sds, is based on the theory of Hasselmann (1974), tuned according to Janssen (1989) and Janssen and Günter (1992). • , and m are constants. In WAM cycle 4 the values for , and m are respectively, 4.1x10-5, 0.5 and 4. In the present implementation the tunable constants are and while m = 4. The default values for and are respectively, 4.5 and 0.5.

22. Source terms (FS) • Bottom friction底摩阻 • The rate of dissipation due to bottom friction is given by • 取决于底摩阻的耗散速度为： • where Cf is a friction coefficient. The coefficient Cf is typically in the range 0.001-0.01 m/s depending on the bed and flow conditions. • 系数Cf根据底床和水流条件取为0.001-0.01m/s。

23. Source terms (FS) • Wave breaking波浪破碎 • The formulation of wave breaking is based on the breaking model by Battjes and Janssen (1978) and Eldeberky and Battjes (1995) • where is a calibration constant, is the fraction of breaking waves, is the mean relative frequency, is the total wave energy, and is the maximum wave height. The value of the breaker parameter, , varies from 0.5 to 1.0 • 其中是校准常数，Qb是破碎波份数，是平均频率，是总波能， • 是最大波高， 破碎参数，0.5-1.0

24. Source terms (DS) • Wind (only in stationary formulation) • 风（只在准定常公式） • The source terms is obtained as the time derivative of the following expressions • Where ~ indicate dimensionless values (wind speed and • gravitational acceleration) and E1 and Ω1 are thetotal wave • energy and the overall mean frequency

25. Space discretization • Cell-centered finite volume method中心有限体积法 • - subdivision of the continuum into non-overlapping cells/elements • 将连续区域细分为不重叠的单元 • - Ni are average values over the ith cell and stored • at the cell center • Ni是第i个网格上的 平均值，存在网格中心 • Geographical domain • - unstructured mesh • - triangles or quadrilateral elements

26. Space discretization • Directional domain • - equidistant (360 degree rose) • - equidistant (directional sector) • Frequency domain (FS) • - equidistant • - logaritmic

27. Space discretization Conservation equation for at scalar quantity U Integrating over the ith cell and using Gauss’s theorem to rewrite the flux integralgives where Ai is the area of the cell, Ωis the integration variable defined on Ai, is the boundary of the ith cell and ds is the integration variable along the boundary. n is the unit outward normal vector along the boundary

28. Space discretization Evaluating the area integrals by a one-point quadrature rule, the quadrature point being the centroid of the cell, and evaluating the boundary integral using a mid-point quadrature rule, the following equation is obtained Here Ui and Si, respectively, are average values of U and S over the ith cell and stored at the cell center, NS is the number of sides of the cell, Fjis the flux vector at the midt point of the jth side, njis the unit outward normal vector at the jth side is the length of the jth interface

29. Space discretization • Integrating over the area of the ith element, the frequency increment and the directional increment • where is the convective flux • where is the normal flux through the edge p with length in the geographical space

30. Space discretization • Convective flux calculation • - first-order upwinding scheme • where is the propagation speed normal to the cell face

31. Time integration • Fractional step approach • Propagation step (Multisequence integration scheme) • 传播步长 • - explicit Euler scheme精确的欧拉格式 • - time step is limited by the local CFL condition时间步长由CFL条件限制 • - maximum local time step

32. Time integration - local time step - the time step index, g, is determind as the minimum value for which - two neighbouring elements are not allowed to have an index difference greater than one两个相连的单元不允许有指数差大于1 - the edges get the lowest index of the two elements they support 边缘指数是两个单元的最小值 - the calculation is performed using a group concept in that groups of elements are identified by their index, g 计算用一组概念完成，在该组中单元由指数g确定。

33. Time integration • Source term step源项时间步长 • - implicit method精确的方法 • Using a Taylor series to approximate Sn+1 and assuming the off- • diagonal terms in the functional derivative to be negligible • such that the diagonal part , • growing waves (>0) =0 (explicit forward difference) • decaying waves (<0) =1 (implicit backward difference) • 对于生长的波浪(>0)，采用向前差分(=0 )，对于衰减的波浪(<0)，采用向后差分(=1)

34. Time integration - limiter The limiter proposed by Hersbach and Janssen (1999) is applied where max is the maximum discrete frequency and defined by Here is the wind friction speed.

35. New stationary solver for SW Conservation equation for wave action where N(x, y, σ, θ)is the action density (x,y) is the Cartesian co-ordinates, (cx, cy,, cσ,cθ)is the propagation velocity of a wave group inthe four-dimensional phase space x, y, σ and θ Sis the source term for the energy balanceequation

36. New stationary solver for SW • For k = 0,1,2 .. Number of iterations do: • stop when and • is obtained from the convective terms in geographical space and • the diagonal part of the functional derivative of the source term • is a relaxation factor, 0 < α≤ 1

37. New stationary solver for SW • Root-mean-square norm均方根标准 • Max norm最大标准

38. New stationary solver for SW • Marching procedure for solving the linear set of equations in each • iteration step

39. New stationary solver for SW • Output Facilities • Overall information of the convergence (dfs0 file) • 输出全部收敛信息（dfs0文件） • Time step number时间步长 • Iteration number迭代数 • RMS-norm of the change in significant wave height between two iterations steps两个迭代步之间有效波高的标准均方根的变化 • Max-norm of the change in significant wave height between two iterations steps两个迭代步之间有效波高的最大值的变化 • RMS-norm of the residual (only method 1)剩余向量的标准均方根（方法1） • Time step interval (only method 2)时间步长（方法2） • Significant wave height in selected points采样点的有效波高

40. New stationary solver for SW • Output Facilities • Domain information of the convergence (dfsu file) • 输出全域的收敛信息（dfsu文件） • Significant wave height 有效波高 • Mean wave period 平均波周期 • Mean wave direction 平均波向 • Change in significant wave height between two iterations steps • 两个迭代步之间的有效波高变化 • RMS-norm of the local residual (spectral domain) • 全域剩余量（谱空间）的标准均方根 • Can be stored after each quasi-stationary time step or with a user • defined frequency • 用户选择中间输出需要指定输出文件名和储存信息的频率

41. New stationary solver for SW Example – Gellen Bay Decoupled parametric formulation Number of elements: 43016 Maximum edge length: 350 m Minimum edge length: 20 m (m)

42. New stationary solver for SW

43. New stationary solver for SW • Iteration in the time domain (old method) • 时间域的迭代 Point 2 Point 1

44. New stationary solver • Newton-Raphson iteration (new method) • 牛顿—拉富生迭代法 α=0.5 α=0.25 α=0.1 α=0.5 α=0.25 α=0.1

45. New stationary solver • Newton-Raphson iteration (new method) α=0.1 α=0.25 α=0.5 α=0.5 α=0.25 α=0.1

46. New stationary solver • Performance执行 • CPU time for pre-processing for new solver: 209 s

47. Dynamic time step动态的时间步长 • Overall ├─────────┼─────────┼─────────┼─────────┤ • Hydrodynamic ├────┼────┼─┼─┼─┼─┼─┼┼┼┼┼┼──┼─┼────┼────┤ • Advection-dispersion├─────────┼─┼─┼───┼─┼─┼──┼────┼─────────┤ • Spectral Waves ├─────────┼────┼────┼────┼────┼─────────┤ • Process description ├───────────────────┼───────────────────┤ • Morphological update ├─────────┼─────────┼─────────┼─────────┤ • time • The time step for the Hydrodynamic module and the Spectral Wave module is determined to satisfy the stability restrictions 水动力和SW模型的时间步长要满足稳定要求 • All time steps are synchronized at the overall time step • HD and AD time step are synchronized at AD time step