# VidConv: A modernized 2D ConvNet for Efficient Video Recognition

Chuong H. Nguyen, Su Huynh, Vinh Nguyen, Ngoc Nguyen
CyberCore AI, Ho Chi Minh, Viet Nam
<EMAIL_ADDRESS>
###### Abstract

Since being introduced in 2020, Vision Transformers (ViT) has been steadily
breaking the record for many vision tasks and are often described as “all-you-
need” to replace ConvNet. Despite that, ViTs are generally computational,
memory-consuming, and unfriendly for embedded devices. In addition, recent
research shows that standard ConvNet if redesigned and trained appropriately
can compete favorably with ViT in terms of accuracy and scalability. In this
paper, we adopt the modernized structure of ConvNet to design a new backbone
for action recognition. Particularly, our main target is to serve for
industrial product deployment, such as FPGA boards in which only standard
operations are supported. Therefore, our network simply consists of 2D
convolutions, without using any 3D convolution, long-range attention plugin,
or Transformer blocks. While being trained with much fewer epochs (5x-10x),
our backbone surpasses the methods using (2+1)D and 3D convolution, and
achieve comparable results with ViT on two benchmark datasets.

##  1 Introduction

Efficient video recognition models are desired for many applications, such as
security monitoring, shop-lifting protection, or driver-action recognition,
where the trade-off between efficiency and accuracy is critical. Despite
recent advances in deep network architecture, selecting a model that can run
on embedded devices and achieve high accuracy for video recognition is still
challenging. The main building blocks of current methods rely either on 3D
convolution (3D ConvNet) or Transformer. They achieve state-of-the-art (SOTA)
accuracy by gradually stacking more layers or plugins and being training with
lots of extra data. Unfortunately, the price for a little accuracy improvement
is often an expensive and much longer training. In addition, while adding
novel operators can be easily implemented with popular framework such as
TensorFlow and Pytorch on GPU device, optimizing parallel computation unit
(PCU) of embedded devices for these operators can be very costly and time-
consuming, thus rarely supported in commercial devices.

(a) Accuracy _vs_. FLOPs

(b) Accuracy _vs_. Training Time

Figure 1: Comparision of our prosposed VidConv (ViC) with SOTA approaches for
action recognition on Kinetics-400. FLOPs is reported for total number of
views (more details in Table 3).

In this paper, we aim to bridge the gap between achieving good accuracy and
being efficient for training and deployment by introducing a new backbone
network for video recognition, dubbed as VidConv:

  * •

Our model uses only standard 2D convolution that is fully supported in all
embedded devices and deployment library. In particular, we neither inflate a
2D Convolution into 3D version, nor adding self-attention as in Transformer.
Instead, we flatten a 3D video tensor $T\times H\times W$ into a 2D tensor
$hH\times wW$, where $T=h.w$ is the number of frames in a clip, and $H,W$ are
the height and width of a frame, respectively. Then, we use two 2D depth-wise
convolutions: one with $7\times 7$ kernel and another of kernel $(h,w)$ and
dilation $(H,W)$ to aggregate spatial and temporal features. This simple
design reduces the number of parameters significantly relative to 3D
counterpart, and does not consume large memory and computational softmax
operator as in attention module.

  * •

As shown in Fig. 1(b), our model needs much less training time
(5$\times$-10$\times$), thus it is much more cost-effective and suitable for
industrial applications.

  * •

Efficiency - Accuracy trade off: Since using only 2D convolutions, our model
is simple and very efficient. On two benchmark datasets Kinetic-400 and
Something-Something, our models surpass all recent methods using (2+1)D or 3D
Convolution, and achieve comparable results with Vision Transformer models.

##  2 Related Work

###  2.1 CNN and variant

The dominant approach for video understanding has been based on 3D Convolution
Neural Networks (3D-CNN). Among them, Slow-Fast Network [14] has been
considered as the strong baseline for action recognition and other downstream
tasks. However, 3D CNN suffer from the large computation cost and difficult
optimization. The early works, such as I3D [3], constructs the 3D CNN by
inflating the pretrained 2D convolution models, which help to stabilize the
training due to better initialization. To reduce the complexity, later works
([35], [40], [34]) factorize the 3D kernel into (2+1)D convolution in the
spatial and temporal dimensions. X3D [13] and MoviNet [18] are developed by
Neural Architecture Search (NAS) to optimize the model efficiency. However,
convolution networks are recently considered inferior to Transformer-based
approaches due to its limited receptive field, thus can’t capture the long
range temporal information.

###  2.2 Temporal Module plugin

Other methods design plugin-modules that are inserted into middle layers of
the standard 2D networks, such as ResNet, to capture short-term and long-term
temporal information. Non-local Network [39] was proposed to improve the
ability to model long range relationship between the frames. Temporal Shift
Module [23] shifts a portion of channels along the temporal axis to fuse the
information between the frames. TAM [27] propose a two-level module, which
decouples the dynamic kernel into a location sensitive map and a location
invariant aggregation weight. The former is learned in a local temporal window
to capture short-term information, while the later is generated from a global
view with a focus on long-term structure. Kwon _et al_. [19] propose the
Motion Squeeze module that extracts motion cues using frame-wise appearance
correlation, and combines the motion feature into the next downstream layer.
Similarity, STM [17] and TEA [22] encode motion information by differentiating
adjacent features across frames, which are then combined with the spatial
features. However, these methods are convolved along the spatial and temporal
axes separately. This limits the interaction between spatial and temporal
features compared to 3D spatio-temporal or transformer based methods. Hence,
Lee _et al_. [20] propose a temporal 3D module that aggregates hierarchy
features with different temporal receptive fields. This helps capture the
large visual tempo variation of the action. Although these temporal modules
can boost the performance of the networks, they often intertwine different
branches and connections across the layers and frames, limiting it for
deployment on hardware devices.

###  2.3 Vision Transfomer and variant

Vision Transformer (ViT) ([10], [33], [24]) has been creating a new wave in
network architecture design for vision tasks. Motivated from the success of
transformer in NLP, ViT divides an image into many patches, which is analogous
to visual words and then applies the vanilla attention mechanism [37] on these
patches to extract the features. Extending the “2D-patchifying” approach into
3D domain for video, several recent works successively propose different
variants for spatiotemporal learning, such as VTN [31], STAM [30], TimeSformer
[1], Video Swin-Transformer [26] and MultiScale ViT [11]. Self-Attention
mechanism is believed to be the key for success of ViT thanks to its ability
to capture the long-range relationship. However, ViT models are often memory
consuming and computational. In addition, recent research [25], [41], [32]
show that simple MLP and Convolution networks, if trained in a similar recipe,
can compete favorably with ViT, questioning the true effectiveness of ViT.

Therefore, in this work, our objective is not to introduce a new or complex
temporal module. Instead, we select a minimal design of the 2D ConvNet, that
is typically considered inferior to 3D ConvNet or ViT for action recognition,
and prove that its performance is still very competitive while preserves the
efficiency. We hope that our model can serve as a simple and strong baseline,
and offer a practical solution for product deployment.

##  3 VidConv Model

###  3.1 Overall Architecture

Our VidConv is built upon the recent modernized ConvNet, particularly the
ConvNeXt [25] thanks to its high performance and simplicity, but other
variants such as PoolFormer [41] are still applicable.

Figure 2: Illustration of VidConv Network with $L=9$ frames. We stack low
level features from 3D dimension $(L\times H_{1}\times W_{1})$ into 2D feature
of spatial dimension $(hH_{1}\times wW_{1})$, where $h=w=3$,
$(H_{1},W_{1})=(H/8,W/8)$ and $(H_{2},W_{2})=(H/32,W/32)$.

(a) VidConv Block Diagram

(b) Visualize VidConv Operation

Figure 3: VidConv block diagram and its operation. MLP is Multi-Layer-
Perceptron.

Figure 2 illustrates the overall architect of VidConv model. The VidConv takes
as input a clip $X\in\mathbb{R}^{L\times H\times W\times 3}$ consisting of $L$
RGB frames of size $H\times W$ sampled from the original video. The network
has a stem layer and 4 stages, after which the feature is down-sampled by a
half. The network’s stages can then be split into two parts.

The early-net includes the stem layer, stage 1st and 2nd as in the original 2D
ConvNeXt [25]. It processes a batch of $L$ frames as regular 2D images, and
outputs $L$ features of size $H/8\times W/8\times C$.

However, before passing the feature into stage 3rd and 4th of the later-net,
we first rearrange the feature of $L=hw$ frames into a grid of $h\times w$,
forming a “collage feature” of size $hH/8\times wW/8\times C_{1}$. For
example, for $L=9$, we arrange the features into $h=3$ vertically and $w=3$
horizontally. The later-net then processes the collage feature as it is
extracted from a single 2D image, and output a tensor of shape $hH/32\times
wW/32\times C_{2}$.

Intuitively, this design casts the problem from recognizing a video of $L$
frames into classifying a collage image with $h\times w$ higher resolutions.
The motivation is that modern ConvNet and ViT often use large kernel size,
_e.g_. $7\times 7$, and more number blocks in stage 3rd and 4th, _e.g_.
ConvNeXt-small models has 27 blocks in stage 3rd. Hence, flattening the 3D
feature of a video into 2D form is the simplest way to capture temporal
structure by exploiting the large receptive field of networks.

In addition, we introduce the VidConv block in the later-net to increase the
temporal connection between the frames. As illustrated in Fig. 3(a), the
VidConv block is a minimal modification of the ConvNeXt block, in which we add
a depth-wise convolution of kernel size $(h,w)$ and dilation $(H,W)$ without
padding.

Concretely, the VidConv block takes an input feature of size $hH\times
wW\times C$. Then, the depth-wise convolution $7\times 7$ extracts the spatial
features $S\in hH\times wW\times C$, and the dilated (also depth-wise)
convolution extracts the temporal feature $T\in H\times W\times C$. The
spatial-temporal fusion is then obtained by adding $T$ and $S$:

$F=S+\alpha*[T]_{(\times h,\times w)}$ (1)

where $[\cdot]_{(\times h,\times w)}$ denotes repeating the tensor $h$ times
horizontally and $w$ vertically, $\alpha\in\mathbb{R}^{C}$ is the learnable
vector to balance between temporal and spatial branch, and $*$ is the element-
wise product. This design allows sharing the temporal features $T$ for all the
frames. Feature $F$ is then passed to Norm Layer and MLP layers.

In fact, there can be many variants for this design, such as using non-shared
weight convolution, dynamics convolution [4], deformable convolution [8] or
attention blocks [37]. Using depth-wise dilated convolution is one of the most
efficient and simplest options. The design is inspired from our hypothesis
that MLP can be the main factor contributing to success of ViT, since MLP
constitutes more then two-thirds of network’s parameters and it is the only
unit having Nonlinear activation. For the same reason, PoolFormer [41] only
uses Pooling operator to perform spatial connection, or ConvMixer [36] simply
uses depth-wise convolution. We follows this minimalist design.

Finally, the feature extracted from later-net is fed to a neck, which is
implemented as a simple convolution of kernel size $(h,w)$, dilation
$(H_{2},W_{2})$ without padding to reduce the spatial size and aggregate the
temporal features before global pooling.

###  3.2 Architecture Variants and Pre-trained Weights

In this work, we focus on small and medium size models. Our VidConv Model
inherits the configuration of ConvNeXt models [25], and Tab. 1 summarizes the
architecture of these model variants. Figure 4 illustrates the block
configuration of VidConv-Tiny.

Table 1: VidConv Model Configuration.

Model Channel of each stage Number of Blocks $C_{3}$ dim Tiny (ViC-T)
$(96,192,384,768)$ $(3,3,9,3)$ $2304$ Small (ViC-S) $(96,192,384,768)$
$(3,3,27,3)$ $2304$ Base (ViC-B) $(128,256,512,1024)$ $(3,3,27,3)$ $2304$

Figure 4: Block diagram of VidConv-tiny, referred as ViC-T. $B$ denotes for
number of clips, and $T$ for number of sampled frames in a clip, $(H,W)$ is
the image size. $k\times k$ denotes the kernel size, $s$ is the stride of
convolution, and block $\times n$ means stacking $n$ blocks in sequence.

Finally, keeping the configuration similar to the original ConvNeXt allows
using their strong pre-trained ImageNet for network initialization. Hence, we
only need to initialize the dilated convolutions in VidConv blocks from
random.

##  4 Experiments

###  4.1 Datasets

We evaluate VidConv on 3 benchmark action recognition datasets: Kinetics-400
(K400) [3], Kinetics-600 (K600) [2] and Something-Something V2 (SSv2) [15].
Kinetics datasets consist of 10-second videos sampled at 25fps from YouTube.
K400 has about 240k training and 20k validation videos in 400 human action
categories. K600 is the extension of K400, and consists of 370k training
videos and 28.3k validation videos in 600 human action categories. SSv2
contains 220k videos sampled at 12fps with duration ranging from 2 to 6
seconds. The dataset is split to 169k training and 24.7k validation videos
over 174 fine-grain action categories. Follow the prior works, we report the
accuracy Top-1 and Top-5 on the validation set for all methods.

###  4.2 Implementation Details

Our code is implemented using MMAction2 framework [6]. We train the models
with AdamW optimizer [29] with a batch size of 64. We use a cosine decay
learning rate scheduler and linear warm-up in the first epoch. The initial
learning rate is 1e-3, and decays to the minimal value 5e-6. To regularize the
model, we use the stochastic depth with drop path rate 0.25 for ViC-T, 0.4 for
ViC-S and 0.5 for ViC-B models. We apply standard image augmentation including
left-right flip and multi-scale cropping. Because the temporal dilated
convolution is initialized randomly, we set the initial value for $\alpha$ in
(1) to 1e-2. In addition, follow [26], since the head and the final dilated
convolution are initialized randomly, we multiply the learning rate of
backbone by a factor $l_{b}$ to stabilize the training.

For K400 and K600, we use the models pretrained on ImageNet-1k for ViC-T and
ViC-S, and models pretrained on ImageNet-21k for ViC-B for initialization. We
train the models for 24 epochs, and set $l_{b}$ to 0.25 for ViC-T, and 0.1 for
ViC-S and ViC-B models. Unless otherwise mentioned, for all model variants, we
sample a clip of 9 frames from each full length video using a random temporal
stride in range of 8 to 12, and spatial size of $224\times 224$. For
inference, we randomly sample 4 clips from a full video, each clip has 9
frames using a fixed temporal stride of 12. The final score is computed as the
average score over all the views.

For SSv2, follow the previous works, we use the models pretrained on K400 for
initialization. However, we found that the models are prompt to be overfit.
Hence, we only train the tiny and small models for 20 epochs with early-
stopping, and set $l_{b}$ to 0.25 for both ViC-T and ViC-S. We also apply
strong augmentation with RandAugment [7], and drop-out after the global
pooling with ratio 0.65 for ViC-T and 0.8 for ViC-S. Unless otherwise
mentioned, for all model variants, we sample a clip of 9 frames from each full
length video using a random temporal stride $s_{T}$ in range of 5 to 7, and
spatial size of $224\times 224$. For inference, we randomly sample 2 clips
from a full video, each clip has 9 frames using a fixed temporal stride of 6.
For any videos that don’t have enough frames, we select the maximum temporal
stride that can fit entire the videos. Concretely, $s_{T}=\min(5,(N-1)//8)$
where $N$ is the number of frames. The final score is computed as the average
score of 2 clips with 3 random crops.

###  4.3 Comparison to state-of-the-art

####  4.3.1 Notation

In this section, “Views” indicates number temporal clips $\times$ number
spatial crop. For example, the notation of $4\times 3$ views indicates a video
is split into 4 clips with different start indices, and frames in each clip
are cropped into left, center and right portions, yielding total 12 views.
Frames (number), FLOPs (Giga $10^{9}$) and Latency (ms) are reported for a
single view, and the corresponding total amounts are multiplied by the number
of views. The latency is reported on a single GPU Nvidia-2080Ti, and the
parameters are in Mega ($10^{6}$). Except the TimeSformer-HR uses image size
of $448\times 448$, other models use image size of $224\times 224$ or
$256\times 256$. IN-1K and IN-21K denotes pretrained on ImageNet-1k and
ImageNet-21k [9] respectively. Bolt and Bold-Italic indicates the highest and
second highest score ($\pm 0.1$) in each group.

####  4.3.2 Kinetics-400

Table 3 compares our solution on Kinetics-400 to the state-of-the-art (SOTA)
Convolution and Transformer based methods in the first and second part of the
table, respectively. Convolution models typically require much longer training
epochs (100-300 epochs), and more frames and views (1$\times$ 10 to $10\times
3$) for equal performance. For transformer-based models, although having less
number of parameters, their FLOP and latency are typically higher.
TimeSformer-L, Swin-S, Swin-B, Uniform-S achieve accuracy $80.7(\pm 0.1)$
Top-1 and 94.7 Top-5, but trained and inference with more frames per views,
_e.g_. TimeSformer-L uses 96 frames per clip or Swin models are inferred with
12 clips of 32 frames. Our base model ViC-B also achieves 80.2 Top-1 and 94.4
Top-5 accuracy using only 4 clips of 9 frames input, closely match the
performance of TimeSformer and Swin models. Note that, inference speed of ViC
models is significantly faster than ViT-based models because during performing
sliding-window feature extraction, we only need to run the early-net for a new
coming frame, while the low-level feature of the previous frames were already
extracted and can be drawn from a small buffer. This can be achieved because
the early-net process each frame independently.

Table 2: Comparison to state-of-the-art on Kinetics-400.

Method Pretrain Training Epochs Top-1 Top-5 Views Frames /view FLOPs /view
Latency /view Params TSN-R50 [38] IN-1K 100 70.6 89.2 1$\times$10 25 103 31.0
24.3 TANet-R50 [28] IN-1K 100 76.3 92.6 1$\times$3 8 43.1 13.8 25.6 I3D-R50
[3] IN-1K 100 72.68 90.78 10$\times$3 32 43.5 25.7 28.0 SlowFast-8x8 [14] -
196 77 92.6 10$\times$3 32 65.8 38.1 34.5 SlowFast-16x8[14] - 256 78.9 93.5
10$\times$3 32 213 N/A N/A X3D-M [13] - 300 76 92.3 10$\times$3 16 6.2 12.1
3.8 X3D-XL [13] - 300 79.1 93.9 10$\times$3 16 48.4 N/A 11.0 TSM-R50 [23]
IN-1K 100 74.7 - 10$\times$3 16 65.9 22.9 24.3 TimeSformer [1] IN-1K 15 78
93.7 1$\times$3 8 100.9 50.5 121.4 TimeSformer-HR [1] IN-1K 15 79.7 94.4
1$\times$3 16 807.2 403.2 121.4 TimeSformer-L [1] IN-21K 15 80.7 94.7
1$\times$3 96 1210.8 515.4 121.4 Swin-T [26] IN-1K 30 78.8 93.6 4$\times$3 32
88 57.7 28.2 Swin-S [26] IN-1K 30 80.6 94.5 4$\times$3 32 166 93.5 49.8 Swin-B
[26] IN-1K 30 80.6 94.6 4$\times$3 32 282 127.9 88.1 MViT-B 16$\times$4 [11] -
200 78.4 93.5 5$\times$1 16 70.5 28.7 36.6 MViT-B 32$\times$4 [11] - 200 80.2
94.4 5$\times$1 32 170 86.6 36.6 Uniformer-S [21] IN-1K 110 80.8 94.7
4$\times$1 16 167 ?? 21.4 Uniformer-B [21] IN-1K 110 82.0 95.1 4$\times$1 16
389 ?? 49.4 ViC-T [Ours] IN-1K 24 75.4 92.2 4$\times$1 9 40.9 11.4 44.7 ViC-S
[Ours] IN-21K 24 78.2 93.5 4$\times$1 9 79.0 22.4 66.4 ViC-S [Ours] IN-21K 24
79.5 94.0 4$\times$1 16 140.4 78.9 ViC-B [Ours] IN-21K 24 80.2 94.4 4$\times$1
9 139.2 33.8 107.4 ViC-B [Ours] IN-21K 24 80.5 94.5 4$\times$3 9 139.2 33.8
107.4

Table 3: Comparison to state-of-the-art on Kinetics-600.

Method Pretrain Training Epochs Official Report Evaluate on CVDF-K600 Views
Frames /view FLOPs /view Params Top-1 Top 5 Top-1 Top-5 SlowFast 16×8 +NL [14]
- 196 81.8 95.1 - - 10$\times$3 16 234 59.9 X3D-M [13] - 256 78.8 94.5 - -
10$\times$3 16 6.2 3.8 X3D-XL [13] - 512 81.9 95.5 - - 10$\times$3 - 48.4 11
TimeSformer [1] IN-21K 15 79.1 94.4 84.6 96 1$\times$3 8 196 121.4
TimeSformer-HR [1] IN-21K 15 81.8 95.8 88 97.2 1$\times$3 16 1703 121.4
TimeSformer-L [1] IN-21K 15 82.2 95.6 87.9 97 1$\times$3 96 2308 121.4 Swin-B
[26] IN-21K 30 84 96.5 87.2 97.3 4$\times$3 32 282 88.1 MViT-B, 16x4 [11] -
200 82.1 95.7 - - 5$\times$1 16 70.5 36.8 MViT-B, 32x3 [11] - 200 83.4 96.3 -
- 5$\times$1 32 170 36.8 MViT-B-24, 32x3 [11] - 200 83.8 96.3 - - 5$\times$1
32 236 52.9 UniFormer-B IN-1K 200 84 96.4 83.3 93.4 4x1 16 389 49.4
UniFormer-B IN-1K 200 83.8 96.7 84.1 93.6 4x1 32 1036 49.4 ViC-B [Ours] IN-21K
24 - - 86.1 96.2 4$\times$1 9 139.2 107.4

####  4.3.3 Kinetics-600

We select model ViC-B for comparison with other state-of-the-art methods. The
experiment setup and training parameters are similar to K400 presented in Sec.
4.2, except that we reduce the stochastic depth dropout ratio to 0.25 for the
backbone, and the drop-out ratio to 0.2 for the head. Since the dataset K600
is no longer provided in the official website, we downloaded it from the
CVDFoundation [5]. We denote this version as CVDF-K600 due to a slight
mismatch with the original release. Concretely, the validation set of
CVDF-K600 has about 29.7k videos, but only 17k videos are in the official
version. Therefore, we include the results on two validation set versions if
the checkpoints of other methods are available.

As shown in Tab. 3, our model ViC-B achieves 86.1 Top-1 and 96.2 Top-5
accuracy, and performs better than TimeSformer and Uniformer-B on the
CVDF-K600 version. Note that, most of other methods get better score when re-
evaluating than their official report, except for the Uniformer that slightly
change. The TimeSformer-HR,TimeSformer-L and Swin-B gets the highest score
(_e.g_. 88 Top-1) but with much higher FLOPs and more views.

####  4.3.4 Somthing-Something V2

Table 4 compares our approach with the SOTA methods on SS-v2. Compare to
Convolution based methods, our ViC-S model can closely match the performance
of STM [17], MSNet [19] and TEA [22], which have more intricate modules
designed to capture the long-range relationship. TEA [22] model gets the
highest Top-1 accuracy (65.1) in this group but is tested with 5x more views
($10\times 3$). Compared to ViT based methods, our ViC-S model with 9-frames
input has top-1 64.4, higher accuracy than TimeSformer [1] with significantly
less FLOPs. When using 16 frames input, our ViC-S model achieves Top-1
accuracy 65.1, 2% lower than MViT-B[11] model (67.1). Swin-B [26] and
Uniformer [21] get the highest accuracy in this group, but require more frames
and FLOPs.

Table 4: Comparison to state-of-the-art on Something-Something V2.

Method Pretrain Training Epochs Top-1 Top 5 Views Frames /view FLOPs /view
Params SlowFast R101 8$\times$8 [14] K-400 22 63.1 87.6 1$\times$3 64 106 53.3
TSM-R50 [23] K-400 50 63.3 88.2 2$\times$3 16 62 42.9 STM-R50 [17] IN-1K 50
64.2 89.8 10$\times$3 16 66.5 24.0 MSNet-R50 [19] IN-1K 40 64.7 89.4
1$\times$1 16 67 24.6 TANet-R50 [28] IN-1K 50 64.6 89.5 2$\times$3 16 66 -
TEA-R50 [22] IN-21K 50 65.1 89.9 10$\times$3 16 70 - TimeSformer [1] IN-21K 15
59.5 - 1$\times$3 8 101 121.4 TimeSformer-HR [1] IN-21K 15 62.2 - 1$\times$3
16 807.2 121.4 TimeSformer-L [1] IN-21K 15 62.4 - 1$\times$3 64 1210.9 121.4
Swin-B [26] K-400 60 69.6 92.7 1$\times$3 32 282 88.1 MViT-B 16$\times$4 [11]
K-400 100 64.7 89.2 1$\times$3 16 70.5 36.6 MViT-B 32$\times$3 [11] K-400 100
67.1 90.8 1$\times$3 32 170 36.6 Uniformer-S [21] K-400 200 67.7 88.5
1$\times$3 16 125 21.4 Uniformer-B [21] K-400 200 70.4 92.8 1$\times$3 16 290
49.4 ViC-T [Ours] K-400 20 61.3 86.9 1$\times$3 9 40.9 44.2 ViC-S [Ours] K-400
20 64.4 88.8 1$\times$3 9 79.0 65.9 ViC-S [Ours] K-400 20 65.1 89.6 1$\times$3
16 140.4 78.4

####  4.3.5 Training Efficiency

One important criteria to select models for production is the training
efficiency. Although longer training with strong data augmentation and careful
tuning can sometimes improve the performance 1-2% on benchmark dataset, it is
hardly considered in industry due to the cost expense and delaying time. We
therefore compare the real training time between the models using the same
server with configuration of 4 GPUs Quadro RTX 8000 (48G), 256GB RAM and
Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz (20 cores). For all other methods,
the results are reported by using their official released code. The training
time for dataset K400 is shown in Fig. 1(b), and Tab. 5 shows the training
time and memory consumed for K600 dataset. From Tab. 5, we see that for the
same amount training epoch, the more frames in a clip, the longer training
time, and data loading time becomes the main bottleneck. Our method gets the
best compromise between the consumed training resource and the accuracy. It
only needs 46 GPU hours (2 days) to achieve 86.1 top-1 accuracy, while Swin-B
would need 216 GPU hours (9 days) and TimeSformer and UniFormer need 626 hours
(26 days) and 1824 hours (76 days) respectively.

Table 5: Compare the consuming training resource of state-of-the-art methods.

Method Training Epochs Batch size Frames /clip Image Size GPU Memory (GB/card)
Training time (GPU hours) Top-1 TimeSformer 15 64 8 224 10.3 101 84.62
TimeSformer-HR 15 24 16 448 25.4 518 88 TimeSformer-L 15 24 96 224 26.3 626
87.9 Swin-B 30 64 32 224 21.4 216 87.2 UniFormer-B, 16x4 110 16 16 224 25.3
816 83.3 UniFormer-B, 32x4 110 8 32 224 41.6 1824 84.1 ViC-B [Ours] 24 64 9
224 24.2 46 86.1

###  4.4 Ablation Study

####  4.4.1 Default Model and Training Setting

In this section, we investigate different factors affecting the performance of
ViC. We use ViC-Tiny pretrained in ImageNet-1k and 9 frames per input clip as
the default model. Unless otherwise mentioned, we drop the neck, which is the
final dilated convolution in Fig. 2 in order to isolate its effect from the
backbone. We train the model for 24 epochs on K400 and 20 epochs on Sth-Sth-V2
using standard augmentation, including left-right flip and multi-scale
cropping with image size $224\times 224$. The results are reported with
4$\times$ 1 views on Kinetics-400, and $1\times 3$ views on Sth-Sth V2
dataset.

####  4.4.2 Temporal branch and Spatial Stacking

We examine independent effect of the spatial stacking and the temporal branch,
which is realized by the simple dilated-convolution in Fig. 3(a). The result
is reported in Table 6. The 1st row show the simplest baseline, where we just
use the 2D ConvNeXt model on each frame separately, then perform global
pooling to extract feature.

Table 6: Effect of Temporal branch and spatial stacking.

Temporal Branch Spatial Stacking Kinetics-400 Sth-Sth V2 Top-1 Top-5 Top-1
Top-5 None None 72.2 89.6 39.8 72.4 None $3\times 3$ 73.8 90.6 53.2 80.8 Yes
None 73.2 90.2 58.2 84.6 Yes $3\times 3$ 74.5 91.4 60.4 86.7

In the 2nd row, we apply feature stacking strategy but still using the
original ConvNeXt model. In the 3rd row, we use VidConv block in later-net
without stacking the feature along spatial axes. Concretely, only the dilated-
convolution is applied on stacked feature, while the depth-wise conv $7\times
7$ operates on each frame independently. This is equivalent to the design of
(2+1)D convolution, where we apply two depth-wise 3D convolutions $1\times
7\times 7$ and $T\times 1\times 1$ to the input $C\times T\times H\times W$.
Finally, the last row shows our ViC-T using both spatial stacking and VidConv
block. We see that both strategy can improve the performance compared to the
baseline, and their combination can further improve the results significantly.
The temporal branch is especially useful on the SS-v2 dataset. This is because
the action categories in SS-v2 are more temporal dependent than that of
Kinetics-K400.

####  4.4.3 Spatial Resolution

Table 7 compares the accuracy when increasing the number of frames and spatial
stacking resolution.

Table 7: Effect of spatial stacking resolution. Spatial Stacking | Kinetics-400 | Sth-Sth V2
---|---|---
Top-1 | Top-5 | Top-1 | Top-5
None | 73.2 | 90.2 | 58.2 | 84.6
$2\times 2$ | 73.9 | 90.7 | 56.2 | 83.5
$3\times 3$ | 74.5 | 91.4 | 60.4 | 86.7
$4\times 4$ | 75.5 | 92.0 | 62.3 | 87.7

The 1st row is the same as the 3rd row of Table 6, where we use 9 frames per
clip with the temporal branch but without spatial stacking. We denote this
case as 1$\times$1 for later reference. 2$\times$2 is applied when we use only
4 frames and 2 clips (_e.g_. total 8 frames) during training and testing.
3$\times$3 and 4$\times$4 denotes the default configuration with 9 frames and
16 frames per clip, respectively. We see that for the same number of frames,
computation and parameters, 3$\times$3 configuration improves the result
relatively to 1$\times$1\. The performance of 2$\times$2 is better than
1$\times$1 and lower than 3$\times$3 on K600 dataset as expected. However, on
SSv2 dataset, the case 1$\times$1 is surprisingly better than 2$\times$2\. The
reason is that although 1$\times$1 case does not apply spatial stacking, it
extract the temporal feature from 9 frames, while 2$\times$2 cases only use 4
frames. Since on SSv2 dataset are more temporal dependent, the temporal branch
is dominant in this case. The 4$\times$4 configuration consistently achieves
the highest score on both datasets, proving the importance of number of frames
input.

####  4.4.4 Stacking Level

In Tab. 8, we investigate the optimal stage to perform spatial stacking.
Stacking after 4th stage is equivalent to the baseline 2D ConvNeXt (1st row of
Tab. 6). We see that the earlier we perform stacking feature, the better
temporal information is extracted. Thus, the higher accuracy but also more
computation and longer latency.

Table 8: Effect of spatial stacking at different stages. Stage 2nd is the
default as shown in Fig. 4.

Stages Kinetics-400 Sth-Sth V2 FLOPs /view Latency /view Params Top-1 Top-5
Top-1 Top-5 1st 75.2 91.7 61.0 86.9 40.13 13.89 28.20 2nd 74.5 91.4 60.4 86.7
40.13 11.23 28.19 3rd 73.7 90.93 56.9 84.0 40.12 6.77 28.15 4th 72.2 89.6 39.8
72.4 40.12 5.02 28.13

###  4.5 Effect for Temporal Branch

####  4.5.1 Connect to previous works

StNet [16] design the Temporal block by a single 3D Convolution (Conv-3D) of
kernel $3\times 1\times 1$ to connect the neighbor frames. The temporal block
is inserted right after stage 3rd and 4th of ResNet backbone. It then
introduces the Temporal Xception Block, which includes several Conv-3D of
different receptive field and Max-pooling, before passing to a fully-connected
layer. Compared to StNet, our temporal branch is more simple and efficient.

SIFAR [12] also use a simple approach by stacking the input images along the
horizon and vertical axes to create the “super-image”, and then apply the
standard image classification on it, such as ResNet101 or Swin-Transformer. In
this perspective, SIFAR is the closest work with us. However, our design is
more efficient. First, we stack the features after the 2nd stage instead of
input images. This design can save computation in practice since we can reuse
the feature extracted from the early-net during temporal sliding in a long
video. Second, we add the dilated convolution to aggregate the temporal
information across the frames. More important, this temporal information is
computed in global scope and shared between the frames. Finally, SIFAR relies
on Swin Transformer backbone, while ViC is simply utilizes all 2D Convolution
layers.

Table 9: Compare to other close works on Kinetics-400 and Something-Something
V2 dataset

Method Kinetics-400 SS-v2 Top-1 Top-5 Top-1 Top-5 StNet - R50 [16] 69.85 - - -
StNet - R101 [16] 71.4 - - - SIFAR - SwinB - 7 [12] 79.6 94.4 56.7 83.3 SIFAR
- SwinB - 12 [12] 80 94.5 60.1 86.8 ViC-S - 3$\times$3 78.2 93.5 64.4 88.8
ViC-S - 4$\times$4 79.5 94.0 65.1 89.6

##  5 Conclusions

In this work, we adopt the modernized 2D Convolution network for the task of
video action recognition. Our model can compete favorably with other state-of-
the-art Vision Transformer while retaining the simplicity and efficiency of
standard 2D ConvNets. Our objective is not to introduce a new or complex
temporal module. Instead, we challenge the belief that 2D Convolution is
inferior to 3D ConvNet or ViT for action recognition. Therefore, we select a
minimal design of the 2D ConvNet, and prove that its performance is still very
competitive while preserves the efficiency. We hope that our model can serve
as a simple and strong baseline, and offer a practical solution for product
deployment.

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##  Appendix A Does the temporal branch actually learn temporal information?

To answer this question, we evaluate the accuracy when frame’s order is
shuffled. Should the model learn temporal feature, its performance must be
dependent on the frame’s temporal structure and the accuracy will be drop in
this test. Table 10 shows the behavior of VidConv and several other models
under three testing conditions: normal, reverse order and randomly shuffling.
We see that, on Kinetics dataset, the accuracy just slightly drops for all
models when reverse the order. Uniformer and Swin models get about 3% accuracy
drops when the order is shuffled randomly, while there is almost no dropping
for ViC and TimeSformer. On SS-v2 dataset, the effect is more obvious. There
are two tendencies. VidConv, TSM and TimeSformer get the worst performance
when the order of frames is reverse, while Uniformer and Swin drops most
severely if the frame order is random shuffled.

Table 10: Evaluate when the frame order is shuffled. Frame Order | Kinetics K400 | Something-Something V2
---|---|---
ViC-S |  | TSM
---
R50
| TimeS-
---
former
| Uni-
---
Former
Swin-B | ViC-S |  | TSM
---
R50
| TimeS-
---
former
| Uni-
---
Former-S
Swin-B
normal | 78.2 | 71.8 | 78 | 80.8 | 80.6 | 64.4 | 60.6 | 59.5 | 67.7 | 69.6
reverse | 78.1 | 71.7 | 78 | 80.4 | 80.2 | 25.4 | 19.2 | 28.9 | 23.7 | 30.7
random | 78.1 | 70.7 | 78 | 76.9 | 77.6 | 40.2 | 25.0 | 46.0 | 16.0 | 25.0

We visualize the attention map of VidConv model using CAM [42] in Appendix.

|
---|---
|
|
|
|

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 5: Visualize the attention areas by Class Activation Map (CAM).


# Two-dimensional turbulence in a finite box

I.V. Kolokolov1,2, V.V. Lebedev1,2 1Landau Institute for Theoretical Physics,
RAS,
142432, Chernogolovka, Moscow region, Russia;
2National Research University Higher School of Economics,
101000, Myasnitskaya ul. 20, Moscow, Russia.

###### Abstract

We present theory of two-dimensional turbulence excited by an external force
in thin fluid films on scales larger than the film thickness. The principal
feature of two-dimensional turbulence is the tendency of producing motions of
larger and larger scales thanks to the nonlinear interaction. The tendency
leads to formation of the so-called inverse cascade and, at some conditions,
of big coherent vortices. We discuss the mean velocity profile of the coherent
vortices and the flow fluctuations on the background of the mean velocity for
different regimes. We demonstrate that the regime of strongly interacting
fluctuations leads to an anisotropic scaling inside the coherent vortices.

##  I Introduction

It is a great honor for us to present this work for a collection of articles
dedicated to the 130th anniversary of P.L. Kapitsa. Pyotr Leonidovich’s
discovery of superfluidity significantly expanded our notion of possible
hydrodynamic effects. Hydrodynamics still continues to amaze us with the
variety of phenomena related to it. In this article we present the current
state of the theory of two-dimensional turbulence, which has been actively
developing in recent years.

We consider turbulent states of thin fluid films characterized by an intense
chaotic motions of the fluid. On scales larger than the film thickness, the
hydrodynamic motions in such films can be treated as two-dimensional BE12 .
Other systems where effectively two-dimensional hydrodynamic motions can be
produced are suspended soap or smectic films kellay2002two ; Yablonskii2017 .
Note also the effectively two-dimensional subsystem of the fast rotating
three-dimensional fluid Proudman ; Pedlovsky . The turbulent states in the
above systems differ essentially from the three-dimensional turbulence.

Turbulence can be excited by a permanently acting force or evolve without
external forcing. The first case is called forced turbulence and the second
case is called decaying turbulence. The decaying case is in many aspects
analogous to the pumping case. However, the decaying case has some
peculiarities, that lie without scope of our paper. We examine the forced
case. In our setup, the external force can be static or can be chaotic
function of time. In the second case we assume, that the force has stationary
statistical properties. In addition, one can consider a force periodically
varying in time. In all the cases the turbulent state, being a state with
strong fluctuations, has stationary statistical properties.

Already first theoretical works 67Kra ; 68Lei ; 69Bat  devoted to two-
dimensional turbulence reveal its principal difference from three-dimensional
one. From the theoretical point of view, the difference is related to
existence of two quadratic positively defined quantities (energy and
enstrophy) conserved by two-dimensional Euler equation. That leads to forming
two different cascades produced by the non-linear interaction: enstrophy is
carried from the pumping scale to smaller scales (direct cascade) whereas
energy is carried to larger scales (inverse cascade). The enstrophy is
dissipated due to viscosity at scales smaller than the pumping length and the
energy is dissipated due to bottom friction at scales larger than the pumping
length.

Statistical properties of the velocity fluctuations in the inverse cascade
were investigated both experimentally Tabeling02  and numerically BCV .
Results of the works are in a good agreement with the analytical theory
developed for an unbounded system KraM . Amazingly, the normal Kolmogorov
scaling is observed in the inverse cascade BCV . The normal scaling in $2d$ is
in contrast with the anomalous scaling observed in three-dimensional
turbulence Frisch . The direct (enstrophy) cascade is characterized by its own
scaling laws 71Kra ; 94FL .

For a three-dimensional flow in a fluid, the only dissipation mechanism of the
kinetic energy is viscosity. Thus, there is the only dimensionless parameter,
Reynolds number, characterizing the degree of non-linearity for an
incompressible flow. In contrast, at considering effectively two-dimensional
thin fluid films one deals with two dissipative mechanisms: viscosity and
bottom friction. Of course, bottom friction is reduced to the same viscosity,
however, acting at the scales of the order of the film thickness. Therefore at
the two-dimensional analysis bottom friction has to be taken into account
separately. An interplay of the dissipative mechanisms leads to a more
complicated characterization of the flow nonlinearity in two-dimensional
turbulence.

The traditional way to produce flow in $2d$ hydrodynamics is applying to the
fluid an external periodic in space static force (Kolmogorov force). Such
setup is used both in laboratory experiments with thin liquid films Sommeria86
; Tabeling02 ; Shats09 ; Orlov17  and in numerical simulations MCH04 ;
Mishra15 ; DFKL22 . Then the transition to turbulence appears to be
complicated, it could be soft transition or jump (depending on the ratio
between the pumping length and the box size) and could go through some
intermediate stages. Another possibility, realized in many numerical
simulations, is the random force short correlated in time BCV ; Bof07 ;
Laurie2014 ; frishman17 ; Parfenyev2022 ; Xu .

For an intensive pumping, the inverse cascade, carrying the energy upscales,
leads to an accumulation of the energy at the box size. The accumulation can
lead to formation of coherent vortices of size comparable with the box size.
The coherent vortices are long-living and possess well determined mean
velocity profile. As numerics and experiment show, the profile is isotropic
and corresponds to a differential rotation. As it was first demonstrated in
the work Laurie2014 , where the model of short correlated in time pumping
force was numerically examined, the velocity profile is flat, that is the
(polar) velocity of the profile is independent of the distance to the center
of the vortex. Further the observation was confirmed numerically Parfenyev2022
and experimentally Orlov17  and some theoretical argument in favor of the flat
profile were formulated KL15 ; kolokolov2016structure ; kolokolov2016velocity
; KL17 . Conditions needed for appearing the coherent vortices are discussed
in Ref. DFKL22 .

The plan of our subsequent text is as follows. In Section II we present basic
relations concerning two-dimensional hydrodynamic flows. In Section III we
observe properties of two-dimensional turbulence in an unbounded fluid film.
In Section IV we introduce equations describing the coherent vortex. In
Section V we analyze fluctuations inside the coherent vortex in the so-called
quasilinear regime. In Section VI we examine the regime of strongly
interacting fluctuations inside the coherent vortex. In Section VII we outline
our results and discuss possible perspectives and relations to other physical
phenomena.

##  II General relations

A two-dimensional fluid flow is described by its (two-dimensional) velocity
field $\bm{v}$, it is a function of time and two coordinates. We assume that
the flow is incompressible: $\nabla\bm{v}=0$. The property implies that the
Mach number $v/c$ is small, where $c$ is the speed of sound. This condition is
usually satisfied well in really excited turbulent flows. In the section we
formulate basic equations describing two-dimensional incompressible
hydrodynamics.

Having in mind dynamics of thin films, one uses two-dimensional Navier-Stokes
equation, supplemented by the term related to bottom friction:

$\partial_{t}\bm{v}+(\bm{v}\nabla)\bm{v}+\nabla
p=-\alpha\bm{v}+\nu\nabla^{2}\bm{v}+\bm{f}.$ (1)

Here $\bm{f}$ is the external force (per unit mass), $p$ is pressure (per unit
mass), $\nu$ is the kinematic viscosity coefficient and $\alpha$ is the bottom
friction coefficient. Taking divergence of the equation (1) and using the
incompressibility condition $\nabla\bm{v}=0$, we arrive at the following
relation for the pressure field $p$:

$\nabla^{2}p=-(\partial_{\lambda}v_{\mu})(\partial_{\mu}v_{\lambda}),$ (2)

where Greek subscripts, running over $1,2$, designate vector components along
two orthogonal axes. At deriving Eq. (2) we implied the condition
$\nabla\bm{f}=0$.

We assume that the pumping force $\bm{f}$ has the characteristic wave vector
$k_{f}$. Then $\nu k_{f}^{2}$ is the viscous damping at the pumping scale. It
is instructive to compare the damping with $\alpha$. For thin liquid films it
is easy to achieve the inequality $\alpha\gg\nu k_{f}^{2}$, since $\alpha$ can
be estimated as $\nu h^{-2}$, where $h$ is the thickness of the film. The
characteristic length of the pumping force should be larger than $h$
(otherwise the flow cannot be treated as two-dimensional one), thus
$k_{f}h\gtrsim 1$. That leads to the conclusion $\alpha\gtrsim\nu k_{f}^{2}$.

However, to observe effects related, say, to intensive large-scale vortices in
the turbulent regime, one should make $\alpha$ as small as possible. To
achieve the goal some experimental tricks are used related to multilayered
films Tabeling02 ; Shats09 ; Orlov17 . Note also, that $\alpha$ is small
compared to $\nu k_{f}^{2}$ for the two-dimensional subsystem of the fast
rotating three-dimensional fluid Proudman ; Pedlovsky , where the effective
bottom friction coefficient is related to the convective motion accompaning by
formation of Ekman boundary layers Vergeles21 . Other cases, where small
$\alpha$ can be realized, include suspended soap or smectic films, which have
no contact with a solid boundary kellay2002two ; Yablonskii2017 .

In two dimensions, it is convenient to describe the flow in terms of vorticity
$\varpi$, defined as

$\varpi=\mathrm{curl}\,\bm{v}\equiv\partial_{1}v_{2}-\partial_{2}v_{1}.$ (3)

Obviously, vorticity $\varpi$ is a scalar (or, more precisely, a pseudoscalar)
field. The equation controlling the vorticity field is derived from Eq. (1) by
taking curl of it:

$\partial_{t}\varpi+\bm{v}\nabla\varpi=-\alpha\varpi+\nu\nabla^{2}\varpi+\phi,$
(4)

where $\phi=\mathrm{curl}\,\bm{f}$. Note that pressure $p$ drops from the
equation (4) for vorticity.

To close the equation (4) one should restore the velocity field $\bm{v}$ from
the vorticity field $\varpi$. Due to the incompressibility condition
$\partial_{1}v_{1}+\partial_{2}v_{2}=0$, it is possible to introduce the
stream function $\psi$, related to the velocity components and to the
vorticity as

$v_{1}={\partial_{2}}\psi,\quad
v_{2}=-{\partial_{1}}\psi,\quad\varpi=-\nabla^{2}\psi.$ (5)

To express the stream function in terms of $\varpi$, it is necessary to solve
the Laplace equation $\nabla^{2}\psi=-\varpi$. Generally, the equation should
be solved with suitable boundary conditions. For an unbounded system, one can
exploit the relation

$\psi(\bm{r})=-\frac{1}{2\pi}\int d^{2}x\,\varpi(\bm{x})\ln|\bm{r}-\bm{x}|.$
(6)

After calculating the integral one finds the velocity components in accordance
with Eq. (5), thus expressing the velocity via the vorticity.

For weak external forcing the nonlinear term $(\bm{v}\nabla)\bm{v}$ in the
equation (1) can be neglected and we arrive at the linear equation. In terms
of vorticity, the equation turns to

$(\partial_{t}+\alpha-\nu\nabla^{2})\varpi=\phi.$ (7)

It describes a laminar flow, excited by the external forcing. Due to the
presence of dissipation (viscosity and bottom friction) after some transition
processes the flow becomes stationary, if the force $\bm{f}$ is stationary. If
the force $\bm{f}$ is random with statistics stationary in time then the
velocity $\bm{v}$ in the laminar regime will be random with statistics
stationary in time as well.

An influence of the external force to the flow can be characterized by its
average power (per unit mass) $\epsilon=\langle\bm{f}\bm{v}\rangle$, where
angular brackets mean averaging over time. The quantity $\epsilon$ is often
called the energy flux. Generally, the average $\langle\bm{f}\bm{v}\rangle$
depends on coordinates. We assume that the dependence is smooth, that is its
characteristic length is larger than characteristic scales of turbulent
pulsations. Then the inhomogeneity of $\epsilon$ is irrelevant for the theory.

A simple analysis of the role of the nonlinearity, based on dimension
estimates, shows that there are two dimensionless parameters characterizing
the strength of the hydrodynamic non-linear interaction

$\beta_{\nu}=\frac{\epsilon}{\nu^{3}k_{f}^{4}},\qquad\beta_{\alpha}=\frac{\epsilon
k_{f}^{2}}{\alpha^{3}},$ (8)

where $k_{f}$ is the characteristic wave vector of the pumping force $\bm{f}$.
The first parameter $\beta_{\nu}$ in Eq. (8) is a power of Reynolds number
taken at the pumping scale $k_{f}^{-1}$. The second parameter $\beta_{\alpha}$
in Eq. (8) is related to bottom friction and characterizes its role at the
same pumping scale.

If the inequalities $\beta_{\alpha}\gg 1$ and $\beta_{\nu}\gg 1$ are satisfied
then developed turbulence is excited and pulsations of different scales are
formed due to non-linear interaction of the flow fluctuations. The energy
produced by the forcing at the scale $k_{f}^{-1}$ flows to larger scales
whereas the enstrophy produced by the forcing at the scale $k_{f}^{-1}$ flows
to smaller scales 67Kra ; 68Lei ; 69Bat . Thus two cascades are formed: the
energy cascade (inverse cascade), realized at scales larger than the forcing
scale $k_{f}^{-1}$, and the enstrophy cascade (direct cascade) realized at
scales smaller than the forcing scale $k_{f}^{-1}$.

##  III Turbulence in an unbounded system

Here we consider two-dimensional turbulence in a big box, where its finite
size do not play an essential role. To describe turbulence in the case one can
use the model of an isotropic and homogeneous in space and time turbulence. Of
course, then the pumping force should be isotropic and homogeneous in space
and time as well, at least in the statistical sense.

Multiplying the dynamic equation (1) by $\bm{v}$ and averaging it, one obtains
the energy balance

$\epsilon\equiv\langle\bm{f}\bm{v}\rangle=\alpha\langle
v^{2}\rangle+\nu\langle(\nabla_{\alpha}\bm{v})^{2}\rangle.$ (9)

Remind that the angular brackets mean averaging over time or, in the
theoretical framework, averaging over the system statistics. At deriving Eq.
(9) we omitted all full derivatives over time and coordinates, having in mind
homogeneity in space and time. The relation (9) has simple physical meaning:
the energy pumped into the fluid by the external force is dissipated by bottom
friction and viscosity.

Analogously, multiplying the equation (4) by $\varpi$ and averaging, one
obtains one extra balance equation

$\eta\equiv\langle\phi\varpi\rangle=\alpha\langle\varpi^{2}\rangle+\nu\langle(\nabla\varpi)^{2}\rangle,$
(10)

Again, at deriving Eq. (10) we omitted all full derivatives over time and
coordinates, having in mind homogeneity in space and time. The quantity $\eta$
is often called the enstrophy flux, it can be estimated as $\eta\sim\epsilon
k_{f}^{2}$. The relation (10) has simple physical meaning: the enstrophy
pumped into the fluid is dissipated by bottom friction and viscosity.

One can derive important relations for the energy and the enstrophy cascades
following the Kolmogorov scheme Kol41 ; LL87 . In the case of isotropic
homogeneous in space turbulence one finds for the inverse energy cascade 67Kra
; 68Lei ; 69Bat

$\left\langle\left\\{\frac{\bm{r}}{r}\left[\bm{v}(\bm{r}_{1})-\bm{v}(\bm{r}_{2})\right]\right\\}^{3}\right\rangle=\frac{3}{2}\epsilon
r,$ (11)

where $\bm{r}=\bm{r}_{1}-\bm{r}_{2}$ and angular brackets, as above, mean
averaging over time. At the same conditions one finds for the direct enstrophy
cascade

$\left\langle\frac{\bm{r}}{r}\left[\bm{v}(\bm{r}_{1})-\bm{v}(\bm{r}_{2})\right]\left[\varpi(\bm{r}_{1})-\varpi(\bm{r}_{2})\right]^{2}\right\rangle=-2\eta
r.$ (12)

Here $r\ll k_{f}^{-1}$. Note the opposite signs in the right hand sides of
Eqs. (11,12) reflecting opposite directions of the energy and of the enstrophy
flows.

Based on the relation (11) one can formulate the following estimate for the
velocity differences in the inverse cascade

$|\bm{v}(\bm{r}_{1})-\bm{v}(\bm{r}_{2})|\sim(\epsilon r)^{1/3},$ (13)

where $r=|\bm{r}_{1}-\bm{r}_{2}|$. Validity of the simple estimate (13) is
confirmed by laboratory experiments and numerical simulations Tabeling02 ; BCV
. It is a difference between $2d$ turbulence and $3d$ turbulence, where the
anomalous scaling is observed Frisch . For the direct cascade the estimate for
the velocity differences, based on Eq. 12), reads

$|\bm{v}(\bm{r}_{1})-\bm{v}(\bm{r}_{2})|\sim(\epsilon k_{f}^{2})^{1/3}r.$ (14)

The estimate (14) is correct upto logarithmic corrections 71Kra ; 94FL .

Comparing the nonlinear term and the term with the bottom friction in Eq. (1),
we find from Eq. (13) that the energy cascade is terminated by the bottom
friction at the scale $\sim L_{\alpha}$,

$L_{\alpha}=\epsilon^{1/2}\alpha^{-3/2}.$ (15)

Analogously, comparing the nonlinear term and the viscous one in Eq. (1) and
using the estimate (14) we conclude that the enstrophy cascade is terminated
by viscosity at the scale $\sim L_{\nu}$,

$L_{\nu}=\nu^{1/2}\eta^{-1/6}\sim\nu^{1/2}(\epsilon k_{f}^{2})^{-1/6}.$ (16)

The inequalities $\beta_{\alpha}\gg 1$, $\beta_{\nu}\gg 1$ are equivalent to
the inequalities $k_{f}L_{\alpha}\gg 1$ and $k_{f}L_{\nu}\ll 1$. Thus there
are regions above and below the pumping scale $k_{f}^{-1}$, where the inverse
and the direct cascades are realized.

In accordance with the estimates (11,12) the velocity gradient is independent
of the scale in the direct cascade and diminishes as the scale increases in
the inverse cascade. Besides, the velocity fluctuations increase as the scale
increases in the inverse cascade. That is why in the energy balance (9) the
second term in the right hand side is negligible, it is explained by the
condition $k_{f}L_{\alpha}\gg 1$. By other words, energy is dissipated mainly
by bottom friction at the scale $L_{\alpha}$. Inversely, enstrophy is
dissipated mainly by viscosity at the small scale $L_{\nu}$. Therefore the
first term in the right hand side of Eq. (10) is negligible, it is explained
by the condition $k_{f}L_{\nu}\ll 1$.

Let us formulate the criteria of the applicability of the relations given in
this section. We established that the turbulent pulsations have scales between
$L_{\alpha}$ and $L_{\nu}$. Thus, for validity of the model of the unbounded
system, the box size $L$ should be much larger than the maximal size of the
flow fluctuations $L_{\alpha}$ (15), $L\gg L_{\alpha}$. And the film thickness
$h$ should be much less than the viscous scale $L_{\nu}$ (16), $h\ll L_{\nu}$.

##  IV Coherent vortex

Now we turn to the case of large $L_{\alpha}$, $L_{\alpha}\gg L$. The
condition can be achieved, say, by increasing the force power $\epsilon$, see
Eq. (15). Besides, we assume, that the box size $L$ is still much larger than
the pumping length, $k_{f}L\gg 1$. Then there is the room for the inverse
cascade carrying the energy from the pumping scale $k_{f}^{-1}$ to the scale
$L$. However, the inverse cascade is essentially deformed in the case
$L_{\alpha}\gg L$ in comparison with the unbounded system.

Since the energy, produced by the pumping force, cannot flow to scales larger
than $L$, the energy flux is stopped there and the energy is accumulated until
the incoming energy flux is compensated by the bottom friction. Then we arrive
at the estimate $V\sim\sqrt{\epsilon/\alpha}$ for the velocity fluctuations at
the scale $L$, following from the energy balance (9). The quantity
$\sqrt{\epsilon/\alpha}$ is much larger than the Kolmogorov estimate
$(\epsilon L)^{1/3}$ (13), since such velocity amplitude is characteristic of
the scale $L_{\alpha}$ for the unbounded system, and $L_{\alpha}\gg L$.

The large-scale fluctuations can be chaotic, then we deal with the random
large-scale motion DFKL22 . However, at some conditions coherent vortices
appear that are long-living structures of the size of the order of the box
size $L$. Such coherent vortices were observed both in laboratory experiments
Shats09 ; Orlov17  and in numerical simulations Chertkov2007 ; Laurie2014 ;
Parfenyev2022 . The coherent vortex possesses a well-defined mean velocity
profile defined in the reference system attached to the vortex center. Note
that the center of the vortex moves with some random velocity.

Based on the experimental and the numerical results, one can assert that the
vortex is isotropic in average. By other words, one deals with a differential
rotation, described by the polar velocity $U(r)$ where $r$ is the distance
between the observation point and the center of the vortex. The mean vorticity
of the coherent vortex $\varOmega=\partial_{r}U+U/r$ is a function of the
distance $r$ as well. The quantity $\varOmega(r)$ is maximal at the center of
the coherent vortex.

An equation for $U$ can be derived from the basic equation (1) where one has
to separate the mean flow and fluctuations on its background. Below, we
designate as $\bm{v}$ the velocity of the fluctuations, unlike the previous
sections. Taking the polar component of the equation (1) and averaging it over
time, one finds

$\alpha U=-\left(\partial_{r}+\frac{2}{r}\right)\langle
v_{r}v_{\varphi}\rangle+\nu\left(\partial_{r}^{2}+\frac{1}{r}\partial_{r}-\frac{1}{r^{2}}\right)U,$
(17)

where $v_{r}$ and $v_{\varphi}$ are radial and polar components of the
velocity fluctuations. At deriving Eq. (17) we assumed that the mean pumping
force is zero. The average $\langle v_{r}v_{\varphi}\rangle$ in Eq. (17) is no
other than the off-diagonal component of the Reynolds stress tensor MoYa .

The viscous term in the equation (17) is relevant in the region near the
vortex center, the size of the region can be estimated as $\sqrt{\nu/\alpha}$.
In this region, inside the vortex core, the viscous term dominates, forcing
the solid state rotation $U\propto r$: such dependence turns to zero the
viscous term in the equation (17). Outside the core the viscous term in the
equation (17) is irrelevant. Then the mean profile $U$ is determined by the
balance between the bottom friction and the force, related to the Reynolds
stress tensor, generated by the fluctuations.

Separating in Eq. (1) the mean flow and the fluctuations, one finds the
following equation for the fluctuating velocity

$\displaystyle\partial_{t}v_{\lambda}+{\lfloor(\bm{v}\nabla)v_{\lambda}\rfloor}+\partial_{\lambda}p$
$\displaystyle+\frac{U}{r}\partial_{\varphi}v_{\lambda}-\frac{U}{r}\varepsilon_{\lambda\mu}v_{\mu}-v_{r}\varSigma\frac{\varepsilon_{\lambda\mu}r_{\mu}}{r}$
$\displaystyle=-\alpha v_{\lambda}+\nu\nabla^{2}v_{\lambda}+f_{\lambda},$ (18)

where ${\lfloor A\rfloor=A-\langle A\rangle}$, $\varepsilon_{\lambda\mu}$ is
$2d$ antisymmetric Levi-Civita symbol and

$\varSigma=r\partial_{r}(U/r)=\partial_{r}U-U/r.$ (19)

The quantity (19) is the local shear rate of the mean flow. Now pressure $p$
satisfies the equation

$\displaystyle\nabla^{2}p=-\partial_{\lambda}\partial_{\mu}\lfloor
v_{\mu}v_{\lambda}\rfloor$
$\displaystyle+\frac{2}{r}\left[U\varpi+\varSigma(v_{\varphi}-\partial_{\varphi}v_{r})\right],$
(20)

instead of Eq. (2). Here $\varpi=\mathrm{curl}\,\bm{v}$ – fluctuating
vorticity.

Let us derive an analog of the energy balance (9) for the interior of the
vortex. Multiplying the equation (18) by $\bm{v}$ and averaging, one finds

$\displaystyle\frac{1}{r}\partial_{r}\left[r\langle
v_{r}(p+v^{2}/2)\rangle\right]+\varSigma\langle v_{r}v_{\varphi}\rangle$
$\displaystyle=\epsilon-\alpha\langle
v^{2}\rangle+\nu\langle\bm{v}\nabla^{2}\bm{v}\rangle.$ (21)

Note that the relation is correct even for an inhomogeneous forcing. Due to
the random motions of the vortex the inhomogeneity is effectively averaged.
Therefore in this case $\epsilon$ in Eq. (21) is an average over the box.

The equation for the fluctuating vorticity $\varpi$ inside the coherent vortex
is

$\displaystyle\partial_{t}\varpi+\frac{U}{r}\partial_{\varphi}\varpi+v_{r}\partial_{r}\Omega+\lfloor(\bm{v}\nabla)\varpi\rfloor$
$\displaystyle=-\alpha\varpi+\nu\nabla^{2}\varpi+\phi,$ (22)

where $\varOmega=\partial_{r}U+U/r$ is the mean angular velocity and
$\phi={\mathrm{c}url}\,\bm{f}$, as earlier. The equation (22) can be obtained
by taking $\mathrm{c}url$ of the equation (18) or directly from Eq. (4) by
separating the mean and the fluctuating contributions to the flow velocity.

Let us consider the separations $r$ from the vortex center much larger than
the characteristic length of the fluctuations. Then the term with $\varOmega$
in Eq. (22) is negligible. Passing then to the reference system rotating with
the angular velocity $U(R)/R$ we find from Eq. (22) for $r$, close to $R$

$\displaystyle\partial_{t}\varpi+\frac{\varSigma(R)}{R}(r-R)\partial_{\varphi}\varpi+\lfloor(\bm{v}\nabla)\varpi\rfloor$
$\displaystyle=-\alpha\varpi+\nu\nabla^{2}\varpi+\phi.$ (23)

We conclude that the influence of the mean flow on the flow fluctuations is
reduced to the presence of the effective shear flow.

##  V Fluctuations inside a coherent vortex: quasilinear regime

The mean flow suppresses the flow fluctuations inside the coherent vortex.
That leads to diminishing the non-linear interaction of the fluctuations. We
consider the distances $r$ satisfying $k_{f}r\gg 1$. Then the fluctuations can
be thought as living in the effectively shear flow with the shear rate (19).
As it was demonstrated in Refs. KLT23a ; KLT23b ; KL24  the suppression effect
is relevant, provided

$\varSigma\gg\nu k_{f}^{2},\quad\varSigma^{2}\gg\alpha^{3}/(\nu k_{f}^{2}).$
(24)

The conditions (24) mean that the fluctuations of the size $k_{f}^{-1}$
produced by the pumping force, are essentially deformed by the shear flow
before they are damped by dissipation (viscosity and bottom friction).

The analysis of the interaction corrections to the correlations functions of
the fluctuating vorticity KLT23a ; KL24  shows that the parameter governing
the strength of the interaction of the fluctuations is

$\displaystyle\beta=\frac{\epsilon}{\nu\varSigma^{2}},$ (25)

at the conditions (24). The interaction of the fluctuations is weak provided
$\beta$ is small. Note that the bottom friction coefficient $\alpha$ does not
enter the expression for the perturbation parameter $\beta$ (25) though it
figures in Eq. (24). The parameter (25) can be understood as the ratio of
$\epsilon/(\nu k_{f}^{2})$ and $\varSigma^{2}k_{f}^{2}$. The first factor is
the square of the typical velocity produced by the pumping force in the
presence of viscosity, and the second factor is the square of the velocity
deformation, caused by the shear flow with the rate $\varSigma$ at the pumping
scale.

If the interaction is small then it is possible to neglect the nonlinear term
in the equation (23) to obtain the linear equation for the fluctuating
vorticity $\varpi$. In the literature such situation is called quasilinear
one. In the quasilinear approximation, one can consistently calculate
correlation functions of $\varpi$ solving the equation for $\bm{v}$ in terms
of $\phi$ and averaging the corresponding products of $\varpi$ based on
statistical properties of $\phi$ KL15 ; kolokolov2016structure ;
kolokolov2016velocity ; KL17 ; Frishman18 . One can find then correlation
functions of the velocity. Corrections to the correlation functions related to
the non-linear interaction of the flow fluctuations were examined in Refs.
KLT23b ; KL24 .

Since the nonlinear interaction of the fluctuations is irrelevant in the
quasilinear regime, then the direct and the inverse cascades are absent in the
case inside the coherent vortex. The size of the flow fluctuations, produced
by the pumping force, can be estimated as $k_{f}^{-1}$ in the radial
direction. However, their characteristic size in the angular direction is much
larger since the mean flow elongate the fluctuations in the angular direction.
The degree of the elongation depends on the life time of the fluctuations.
Say, the pair correlation function of vorticity is formed during the time

$\tau_{\star}=\left(\varSigma^{2}\nu k_{f}^{2}\right)^{-1/3}.$

Therefore the characteristic angular size of the fluctuations is

$\varSigma\tau_{\star}k_{f}^{-1}=\left(\frac{\varSigma}{\nu
k_{f}^{2}}\right)^{1/3}k_{f}^{-1}.$

The size is much larger than the radial size thanks to the first condition in
Eq. (24).

One of the results of the calculations in the quasilinear approximation
concerns the correlation function $\langle v_{r}v_{\varphi}\rangle$ figuring
in Eq. (17). As it was demonstrated in Refs. KL15 ; kolokolov2016structure

$\langle v_{r}v_{\varphi}\rangle=\epsilon/\varSigma.$ (26)

Substituting the expression (26) into Eq. (17) we find a closed equation for
the mean velocity $U$, since $\varSigma$ is expressed via $U$ in accordance
with Eq. (19). Outside the viscous core, where the term with the viscosity in
Eq. (17) can be neglected, one finds the solution of the equation

$U=\sqrt{3\epsilon/\alpha},$ (27)

independent of $r$. The flat velocity profile (27) was observed in numerical
simulations Laurie2014 , where the quasilinear regime was held. Note that the
expression (27) is in accordance with the estimate $(\epsilon/\alpha)^{1/2}$
for the large-scale fluctuations found in Section IV.

Calculating the mean vorticity $\varOmega$ corresponding to the flat velocity
profile (27) we find $\varOmega=\sqrt{3\epsilon/\alpha}\,r^{-1}$. Thus the
vorticity grows as $r$ diminishes and at small $r$ becomes much larger than
the typical vorticity of the large-scale fluctuations. One can say that the
coherent vortices accumulate vorticity. The law $\varOmega\propto r^{-1}$
works down to the viscous core where $\varOmega$ is saturated.

One can explain the relation (26) in simple terms based on the energy balance
(21). The main contribution to the average $\langle v_{r}v_{\varphi}\rangle$
is related to the fluctuations with the radial scales of the order of the
pumping length. Thus $\nabla^{2}$ in Eq. (20) is estimated as $k_{f}^{2}$.
Therefore pressure $p$ is small at the condition $k_{f}r\gg 1$ and can be
neglected in Eq. (21). The term, proportional to $v_{r}v^{2}$ in Eq. (21) can
be neglected as well, since it is of third order in $\bm{v}$ and is small in
the quasilinear regime. Outside the viscous core one can neglect the viscous
term in Eq. (21). Since the fluctuations are weak, one can neglect the term
with $\alpha$ in Eq. (21) as well. Thus, one arrives at the relation (26).

In accordance with Eq. (19), the shear rate corresponding to the velocity (27)
is

$\varSigma=-\frac{\sqrt{3\epsilon/\alpha}}{r}.$

Substituting the expression into Eq. (25) one finds

$\beta=\frac{\alpha r^{2}}{3\nu}.$ (28)

The condition of smallness of such $\beta$ is compatible with the inequality
$k_{f}r\gg 1$ only if $\alpha\ll\nu k_{f}^{2}$. The inequality cannot be
achieved in thin fluid films (see Section II), however, it is achievable for
suspended soap or smectic films. There is no problem to satisfy the inequality
$\alpha\ll\nu k_{f}^{2}$ in numerical simulations. The inequality was
satisfied in the simulations reported in Ref. Laurie2014 .

##  VI Fluctuations inside a coherent vortex: strong interaction

Here we consider the case where the coupling constant $\beta$ (25) is large.
Then the nonlinear interaction of the flow fluctuations inside the coherent
vortex becomes relevant. Note that in accordance with the expression (28) the
coupling constant $\beta$ grows as $r$ increases. Therefore one can encounter
the situation where both regimes (the quasilinear one and the regime of strong
interaction) coexist inside a single coherent vortex.

Since the interaction of the fluctuations becomes relevant for large $\beta$,
the direct and the inverse cascades are restored unlike the quasilinear
regime. However, the correlation functions of the velocity and vorticity
characteristic of the cascades can be anisotropic in some regions of scales.
It is a direct consequence of the elongation of the flow fluctuations in the
angular direction produced by the mean flow $U$ of the vortex.

Let us consider the inverse cascade inside the vortex One can extend
Kolmogorov arguments Kol41 ; LL87  for the case. As it follows from Eq. (23),
the effective shear flow is dropped from the generalized relation (11) if
$r_{1}=r_{2}=r$. Therefore we arrive at the estimate

${v_{\varphi}(r,\varphi_{1})-v_{\varphi}(r,\varphi_{2})}\sim(\epsilon
r|\varphi_{1}-\varphi_{2}|)^{1/3},$ (29)

formally coinciding with the isotropic estimate (13).

Comparing the shear rate $\varSigma$ with the nonlinear rate
$r^{-1}\partial_{\varphi}v_{\varphi}$, we find using the expression (29) the
scale

$L_{an}=\epsilon^{1/2}\varSigma^{-3/2},$ (30)

separating the isotropic regime and the anisotropic one. For scales smaller
than $L_{an}$ the mean shear flow is irrelevant and the standard isotropic
inverse cascade is realized whereas for scales larger than $L_{an}$ the mean
shear flow becomes relevant. Thus, for scales larger than $L_{an}$ the inverse
cascade becomes anisotropic. If $k_{f}L_{an}<1$ then the inverse cascade is
anisotropic on all scales above the pumping length.

Let us establish the character of the anisotropy. For the purpose one has to
compare the term with $\varSigma$ and the nonlinear term in the equation (23).
As a result, we find for the second-order structure function

$\langle[v_{\varphi}(r_{1},\varphi_{1})-v_{\varphi}(r_{2},\varphi_{2})]^{2}\rangle=[\epsilon
r(\varphi_{1}-\varphi_{2})]^{2/3}g(\xi),$ (31)

where $r=r_{1}/2+r_{2}/2$, $g$ is some dimensionless function and $\xi$ is the
similarity variable

$\xi=\varSigma(r_{1}-r_{2})[\epsilon r(\varphi_{1}-\varphi_{2})]^{-1/3}.$ (32)

Analogously, higher order structure functions of $v_{\varphi}$ can be
expressed.

The incompessibility condition $\nabla\bm{v}=0$ leads to the conclusion that
$\Delta r\Delta v_{\varphi}\sim r\Delta\varphi\Delta v_{r}$ where we
introduced the differences like in Eqs. (31,32). If the similarity variable
$\xi\sim 1$, then we conclude that $\Delta v_{r}\ll\Delta v_{\varphi}$ due to
$\Delta r\gg L_{an}$, as one could anticipate. Based on the reasoning one can
establish the scaling laws like (31) for the structure functions of any order
containing both, $v_{\varphi},v_{r}$.

Note that if the similarity variable (32) is of order of unity, $\xi\sim 1$,
then

$\frac{\Delta
r}{r\Delta\varphi}\sim\left(\frac{L_{an}}{r\Delta\varphi}\right)^{2/3}.$ (33)

The quantity (33) is small for the region of the anisotropic inverse cascade,
where $r\Delta\varphi\gg L_{an}$. By other words, the characteristic
separation in the angular direction is much larger than in the radial one, as
one expected.

The anisotropic scaling (31) is correct provided $|\varphi_{1}-\varphi_{2}|\ll
1$. If the difference is of order unity then $\xi\sim 1$ leads to

$\Delta r\sim r^{1/3}L_{an}^{2/3}\ll r.$ (34)

If $\Delta r$ becomes larger than the estimate (34) then the term with
$\varSigma$ in the equation (23) overcomes the nonlinear term and we pass to
the quasilinear regime, where the energy cascade is absent. Our notion of the
regime KLT23b ; KL24  enables us to conclude that the quantity (34) is the
correlation length of the flow fluctuations in the radial direction. Therefore
the strongest velocity fluctuations can be estimated as

$v_{\varphi}\sim(\epsilon r)^{1/3},\quad
v_{r}\sim\frac{L_{an}^{2/3}}{r^{2/3}}(\epsilon r)^{1/3},$ (35)

in accordance with Eq. (34).

Now we return to the energy balance (21). As it follows from Eq. (20), the
first contribution to pressure $p$ can be estimated as $\bm{v}^{2}$ where
$\bm{v}$ is the strongest velocity fluctuation determined by Eq. (34). The
second contribution to pressure $p$ related to the last two terms in the right
hand side of Eq. (20) is negligible since Laplacian in this case is estimated
as $(\Delta r)^{-2}$, where $\Delta r$ is determined by Eq. (34). Therefore
the Laplacian produces a big factor at $p$. Thus

$v_{r}(p+v^{2}/2)\sim\frac{L_{an}^{2/3}}{r^{2/3}}\epsilon r,$

in Eq. (21). The derivative $\partial_{r}$ in the term with pressure in Eq.
(21) is estimated as $r^{-1}$ and therefore this term contains the small
factor $(L_{an}/r)^{2/3}$ at $\epsilon$ and can be, consequently, neglected.

Thus we arrive at the same relation (26) and to the same flat velocity profile
(27) as in the quasilinear approximation. The profile is realized provided
there is an anisotropic region of the inverse cascade. Note that for the flat
velocity profile

$L_{an}^{2}\sim r^{3}/L_{\alpha}^{2}\ll r.$

Therefore the expression (27) for the mean velocity is self-consistent.

Now some words about the direct (enstrophy) cascade. If $k_{f}L_{an}\gg 1$
then the direct cascade is the same as in the unbounded system, see Section
III. In the opposite case, where $k_{f}L_{an}\ll 1$ the direct cascade is
anisotropic. The inequality is equivalent to $\varSigma^{3}\gg\eta$.
Analogously to the inverse cascade, $r\Delta\varphi\gg\Delta r$ for the
characteristic values. Thus $\varpi=-\partial_{r}v_{\varphi}$. Comparing then
the term with $\varSigma$ and the nonlinear term in the equation (23) we find
the estimates

$\displaystyle\varpi\sim\varSigma,\quad v_{\varphi}\sim\Sigma\Delta r.$ (36)

As in the unbounded case, the characteristic $\varpi$ is independent of scale.

Expanding the Kolmogorov-Kraichnan reasoning 67Kra ; 68Lei ; 69Bat  to the
anisotropic direct cascade we find the relation

$\langle\Delta v_{\varphi}\varpi\varpi\rangle\sim\eta r\Delta\varphi,$ (37)

for $\Delta r=0$. Substituting the estimates (36) into Eq. (37) we find the
similarity variable

$(\Sigma^{3}\Delta r)/(\eta r\Delta\varphi),$ (38)

controlling correlation functions of vorticity in the direct cascade.

##  VII Conclusion

Two-dimensional turbulence is in some sense reacher than three-dimensional
one. There are two cascades (of energy and of enstrophy) leading to production
of a large diapason of scales of turbulent pulsations. As a consequence of the
energy cascade, there can appear coherent vortices in a finite box. We
examined different regimes of the flow fluctuations in a coherent vortex, that
can be quasilinear or non-linear. In the last case the anisotropic scaling has
to be observed.

The theoretical results presented in the paper are mainly confirmed by
laboratory experiments with thin fluid films and by numerical simulations.
However, the theoretical results concerning the anisotropic scaling in the
non-linear regime wait for their confirmation. The numerical simulations
initiated to check our predictions are now in progress. We are thinking about
experimental check as well.

We considered the simplest model where both, the box and the pumping force are
homogeneous. We believe that the model demonstrates all qualitative features
of two-dimensional turbulence. However one can expand the model to include
into consideration inhomogeneity of the box and of the pumping force. The
results of such model can be in more detail compared to hydrodynamic processes
in environment.

One should note such phenomenon as geostrophic vortices, generated in a
relatively fast rotating fluid, and playing an essential role in geophysics
Pedlovsky . The geostrophic vortices can be described in terms of an
effectively two-dimensional flow governed by the two-dimensional hydrodynamic
equations Ogorodnikov20 . It would be of interest to translate our results
concerning the anisotropic scaling to the geostrophic vortices.

Another possible direction of expansion of our theoretical scheme is passing
to non-Newtonian fluids. Particularly, one can think about polymer solutions.
Effects caused by the elastic degree of freedom related to the polymers could
lead to such remarkable phenomenon as elastic turbulence Steinberg . It would
be of interest to examine peculiarities of elastic turbulence in thin fluid
films.

###### Acknowledgements.

We thank V. Parfenyev for numerous helpful discussions. The work is performed
in the Laboratory “Modern Hydrodynamics” created in frames of Grant
075-15-2022-1099 of the Ministry of Science and Higher Education of the
Russian Federation in Landau Institute for Theoretical Physics of RAS and is
supported by Grant 23-72-30006 of Russian Science Foundation.

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